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Kalkulator Do Całek Nieoznaczonych (Sprawdź, Czy Dobrze Liczysz)

Krystian Karczyński

Założyciel i szef serwisu eTrapez.

Magister matematyki Politechniki Poznańskiej. Korepetytor matematyki z wieloletnim stażem. Twórca Kursów eTrapez, które zdobyły ogromną popularność wśród studentów w całej Polsce.

Mieszka koło Szczecina. Lubi spacery po lesie, plażowanie i piłkę nożną.


Przedstawiam Wolframow’y kalkulator do całek nieoznaczonych, przerobiony troszkę przeze mnie:

Sprawa jest prosta: w kalkulator wpisujemy formułę (zgodnie z Zasadami) – bez dx, klikamy na ‘Oblicz’ i mamy policzoną całkę.

Na przykład, żeby policzyć \int{\frac{{{x}^{2}}}{{{x}^{2}}+1}}dxwpisujemy w kalkulator: x^2/(x^2+1).

Tyle, mam nadzieję, że kalkulator się Tobie przyda. W razie kłopotów z jego korzystaniem, daj znać w komentarzach pod postem.

Jedna z wielu opinii o naszych Kursach...

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  1. pisze:

    Witam, dostałem na kolokwium taką całkę, męczę ją niestety już od tygodnia i nie potrafię jej rozwiązać, czy mógłbym prosić o pomoc? Całka jest następująca: (1 – arctg^9x) *dx / (1 + x^2) * arctgx

    Z góry bardzo dziękuję 🙂

  2. Jassi pisze:

    Mam problem z jednym z zadań, mogę prosić o pomoc ?

    Calka 1/5*cos(x)*sin(x)^2

    Z tego co kojarzę to wynik wynosi 2/15

  3. Asia pisze:

    Mam problem z obliczeniem całki

    1-x^3/x-1

    Proszę o pomoc 🙂

  4. Damian Krynicki pisze:

    Próbuję obliczyć całkę 2x+7/x^3-16x  po dx.

    1. Krystian pisze:

      Sposób rozwiązywania tego typu całek przedstawiłem w moim Kursie Całek Nieoznaczonych.

      Pójdzie tak:

      integral fraction numerator 2 x plus 7 over denominator x cubed minus 16 x end fraction d x equals integral fraction numerator begin display style 2 x plus 7 end style over denominator begin display style x open parentheses x squared minus 16 close parentheses end style end fraction d x equals integral fraction numerator begin display style 2 x plus 7 end style over denominator begin display style x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses end style end fraction d x equals...

      fraction numerator 2 x plus 7 over denominator x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses end fraction equals A over x plus fraction numerator B over denominator x minus 4 end fraction plus fraction numerator C over denominator x plus 4 end fraction space space space divided by times x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses
2 x plus 7 equals A open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses plus B x open parentheses x plus 4 close parentheses plus C x open parentheses x minus 4 close parentheses
2 x plus 7 equals A x squared minus 16 A plus B x squared plus 4 B x plus C x squared minus 4 C x
open curly brackets table row cell 0 equals A plus B plus C end cell row cell 2 equals 4 B minus 4 C end cell row cell 7 equals negative 16 A end cell end table close

      open curly brackets table row cell 0 equals A plus B plus C end cell row cell 2 equals 4 B minus 4 C space space divided by colon 4 end cell row cell A equals negative 7 over 16 end cell end table close
open curly brackets table attributes columnalign left end attributes row cell 0 equals negative 7 over 16 plus B plus C end cell row cell 1 half equals B minus C space space rightwards double arrow space space B equals 1 half plus C end cell end table close
0 equals negative 7 over 16 plus 1 half plus C plus C space space divided by times 16
0 equals negative 7 plus 8 plus 16 C plus 16 C
32 C equals negative 1
C equals negative 1 over 32

      B equals 1 half minus 1 over 32 equals 15 over 32

      ... equals integral fraction numerator begin display style 2 x plus 7 end style over denominator begin display style x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses end style end fraction d x equals integral fraction numerator negative 7 over 16 over denominator x end fraction d x plus integral fraction numerator 15 over 32 over denominator x minus 4 end fraction d x plus integral fraction numerator negative 1 over 32 over denominator x plus 4 end fraction d x equals
equals negative 7 over 16 integral 1 over x d x plus 15 over 32 integral fraction numerator 1 over denominator x minus 4 end fraction d x minus 1 over 32 integral fraction numerator 1 over denominator x plus 4 end fraction d x equals
equals negative 7 over 16 ln open vertical bar x close vertical bar plus 15 over 32 ln open vertical bar x minus 4 close vertical bar minus 1 over 32 ln open vertical bar x plus 4 close vertical bar plus C

      99/38/58957e8897e9f4e5b309cd70cc28.png” alt=”for all for M element of straight real numbers of there exists for N of for all for n greater than N of a subscript n less than M” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo»§#8704;«/mo»«mrow»«mi»M«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«/mrow»«/munder»«munder»«mo»§#8707;«/mo»«mi»N«/mi»«/munder»«munder»«mo»§#8704;«/mo»«mrow»«mi»n«/mi»«mo»§#62;«/mo»«mi»N«/mi»«/mrow»«/munder»«msub»«mi»a«/mi»«mi»n«/mi»«/msub»«mo»§#60;«/mo»«mi»M«/mi»«/math»” />
      Trzeba więc wyjść z nierówności

      a subscript n greater than M

      lub:

      a subscript n less than M

      Ma Pan jakiś konkretny przykład?51/4a/9d54b58ea3e7d8fe464d9a5e5151.png” alt=”f open parentheses x comma y close parentheses equals open parentheses x squared plus y squared close parentheses times e to the power of negative open parentheses x squared plus y squared close parentheses end exponent” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mrow»«/msup»«/math»” /> jest parzysta, to

      A subscript 3 equals A subscript 2 equals 0

      B subscript 3 equals B subscript 2 equals 0

      C subscript 3 equals C subscript 2 equals negative 4 over e

      Dalej liczymy hesjan:

      H equals open vertical bar table row A B row B C end table close vertical bar equals A times C minus B squared

      H subscript 1 equals A subscript 1 times C subscript 1 minus B subscript 1 superscript 2 equals 2 times 2 minus 0 squared equals 4

      H subscript 2 equals A subscript 2 times C subscript 2 minus B subscript 2 superscript 2 equals 0 times open parentheses negative 4 over e close parentheses minus 0 squared equals 0

      H subscript 3 equals H subscript 2 equals 0

      Ponieważ

      H subscript 1 equals 4 greater than 0 comma space A subscript 1 equals 2 greater than 0,

      to w punkcie P subscript 1 open parentheses 0 comma 0 close parentheses funkcja osiąga minimum lokalne, i

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals open parentheses 0 squared plus 0 squared close parentheses times e to the power of negative open parentheses 0 squared plus 0 squared close parentheses end exponent equals 0

      Ponieważ

      H subscript 2 equals H subscript 3 equals 0

      to w punktach P subscript 2 open parentheses 0 comma 1 close parentheses oraz P subscript 3 open parentheses 0 comma negative 1 close parentheses sytuacja jest nieznana (potrzebujemy wiele badań). 

       

      Jednak, jak już mówiono powyżej, współrzędne tych punktów spełniają warunek 

      x squared plus y squared equals 1

      dlatego w tych punktach nie ma ekstrema lokalne.

      Odpowiedź:

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals 0

      5b/4e/4a87bb1fa7976d920fd270009d4b.png” alt=”fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction equals fraction numerator partial differential squared f over denominator partial differential y partial differential x end fraction equals 4″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»x«/mi»«mo»§#8706;«/mo»«mi»y«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»y«/mi»«mo»§#8706;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»4«/mn»«/math»” />

      fraction numerator partial differential squared f over denominator partial differential y squared end fraction equals 12 y squared minus 4

       

      Macierz Hessego ma postać:

      H subscript f left parenthesis x comma y right parenthesis equals open square brackets table row cell 12 x squared minus 4 end cell 4 row 4 cell 12 y squared minus 4 end cell end table close square brackets

       

      H subscript f left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals H subscript f left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals open square brackets table row 20 4 row 4 20 end table close square brackets

      M subscript 1 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 1 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 greater than 0

      M subscript 2 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 2 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 times 20 minus 4 times 4 equals 384 greater than 0

      Zatem w punktach open parentheses square root of 2 comma negative square root of 2 close parentheses oraz open parentheses negative square root of 2 comma square root of 2 close parentheses podana funkcja ma minima lokalne właściwe. 

       

      H subscript f left parenthesis 0 comma 0 right parenthesis equals open square brackets table row cell negative 4 end cell 4 row 4 cell negative 4 end cell end table close square brackets

      M subscript 2 left parenthesis 0 comma 0 right parenthesis equals open parentheses negative 4 close parentheses times open parentheses negative 4 close parentheses minus 4 times 4 equals 0

      Na razie nie wiemy, czy w punkcie open parentheses 0 comma 0 close parentheses funkcja f ma ekstremum. 

      Dla x equals 0 mamy: f left parenthesis 0 comma y right parenthesis equals y to the power of 4 minus 2 y squared. Wtedy punkt open parentheses 0 comma 0 close parentheses to maksimum lokalne funkcji f.

      Dla y equals x mamy:
       f left parenthesis x comma x right parenthesis equals x to the power of 4 plus x to the power of 4 minus 2 x squared plus 4 x squared minus 2 x squared equals 2 x to the power of 4. Wtedy
      punkt open parentheses 0 comma 0 close parentheses to minimum lokalne.

      Zatem w punkcie open parentheses 0 comma 0 close parentheses funkcja f nie posiada ekstremum.

       

      Musimy zbadać jeszcze wartości na brzegach wskazanego obszaru ograniczonego prostymi: x equals 0 comma space y equals 0 comma space x plus y equals 5.

       

  5. Nick pisze:

    Próbuję obliczyć całkę (arcsinx)^2, proszę o pomoc

    1. Krystian Karczyński pisze:

      Pójdzie tak:

      integral open parentheses a r c sin x close parentheses squared d x equals open vertical bar table row cell u open parentheses x close parentheses equals open parentheses a r c sin x close parentheses squared end cell cell v apostrophe open parentheses x close parentheses equals 1 end cell row cell u apostrophe open parentheses x close parentheses equals fraction numerator 2 a r c sin x over denominator square root of 1 minus x squared end root end fraction end cell cell v open parentheses x close parentheses equals x end cell end table close vertical bar equals
equals x open parentheses a r c sin x close parentheses squared minus 2 integral fraction numerator x a r c sin x over denominator square root of 1 minus x squared end root end fraction d x equals open vertical bar table row cell t equals a r c i n x rightwards double arrow sin t equals x end cell row cell d t equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction d x end cell end table close vertical bar equals
equals x open parentheses a r c sin x close parentheses squared minus 2 integral t sin t d t equals open vertical bar table row cell u open parentheses t close parentheses equals t end cell cell v apostrophe open parentheses t close parentheses equals sin t end cell row cell u apostrophe open parentheses t close parentheses equals 1 end cell cell v open parentheses t close parentheses equals negative cos t end cell end table close vertical bar equals
equals x open parentheses a r c sin x close parentheses squared minus 2 open parentheses negative t cos t plus integral cos t d t close parentheses equals x open parentheses a r c sin x close parentheses squared plus 2 t cos t minus 2 sin t plus C equals
equals x open parentheses a r c sin x close parentheses squared plus 2 a r c s i n x times cos open parentheses a r c i n x close parentheses minus 2 sin open parentheses a r c i n x close parentheses plus C equals
equals x open parentheses a r c sin x close parentheses squared plus 2 a r c s i n x times cos open parentheses a r c i n x close parentheses minus 2 x plus C equals

      a1/71/34f3d48205f417ca8f78d7c0342f.png” alt=”integral fraction numerator x plus 9 over denominator x cubed plus 2 x squared plus 3 x end fraction d x equals integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfrac»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/mrow»«mrow»«mi»x«/mi»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mi»d«/mi»«mi»x«/mi»«/math»” />

      Dalej nie da się rozłożyć, bo delta trójmianu kwadratowego jest ujemna:

      integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x equals
increment equals 2 squared minus 4 times 1 times 3 equals 4 minus 12 equals negative 8

      Ułamek rozkładamy na ułamki proste:

      fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction equals A over x plus fraction numerator B x plus C over denominator x squared plus 2 x plus 3 end fraction space space divided by times x open parentheses x squared plus 2 x plus 3 close parentheses
x plus 9 identical to A open parentheses x squared plus 2 x plus 3 close parentheses plus open parentheses B x plus C close parentheses x
x plus 9 identical to A x squared plus 2 A x plus 3 A plus B x squared plus C x
open curly brackets table row cell 0 equals A plus B end cell row cell 1 equals 2 A plus C end cell row cell 9 equals 3 A end cell end table close
9 equals 3 A
A equals 3
open curly brackets table attributes columnalign left end attributes row cell 0 equals 3 plus B end cell row cell 1 equals 2 times 3 plus C end cell end table close
open curly brackets table attributes columnalign left end attributes row cell B equals negative 3 end cell row cell C equals negative 5 end cell end table close
fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction equals 3 over x plus fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction

      Czyli:

      integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x equals integral 3 over x d x plus integral fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction d x

      Cdn. :)99/38/58957e8897e9f4e5b309cd70cc28.png” alt=”for all for M element of straight real numbers of there exists for N of for all for n greater than N of a subscript n less than M” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo»§#8704;«/mo»«mrow»«mi»M«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«/mrow»«/munder»«munder»«mo»§#8707;«/mo»«mi»N«/mi»«/munder»«munder»«mo»§#8704;«/mo»«mrow»«mi»n«/mi»«mo»§#62;«/mo»«mi»N«/mi»«/mrow»«/munder»«msub»«mi»a«/mi»«mi»n«/mi»«/msub»«mo»§#60;«/mo»«mi»M«/mi»«/math»” />
      Trzeba więc wyjść z nierówności

      a subscript n greater than M

      lub:

      a subscript n less than M

      Ma Pan jakiś konkretny przykład?51/4a/9d54b58ea3e7d8fe464d9a5e5151.png” alt=”f open parentheses x comma y close parentheses equals open parentheses x squared plus y squared close parentheses times e to the power of negative open parentheses x squared plus y squared close parentheses end exponent” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mrow»«/msup»«/math»” /> jest parzysta, to

      A subscript 3 equals A subscript 2 equals 0

      B subscript 3 equals B subscript 2 equals 0

      C subscript 3 equals C subscript 2 equals negative 4 over e

      Dalej liczymy hesjan:

      H equals open vertical bar table row A B row B C end table close vertical bar equals A times C minus B squared

      H subscript 1 equals A subscript 1 times C subscript 1 minus B subscript 1 superscript 2 equals 2 times 2 minus 0 squared equals 4

      H subscript 2 equals A subscript 2 times C subscript 2 minus B subscript 2 superscript 2 equals 0 times open parentheses negative 4 over e close parentheses minus 0 squared equals 0

      H subscript 3 equals H subscript 2 equals 0

      Ponieważ

      H subscript 1 equals 4 greater than 0 comma space A subscript 1 equals 2 greater than 0,

      to w punkcie P subscript 1 open parentheses 0 comma 0 close parentheses funkcja osiąga minimum lokalne, i

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals open parentheses 0 squared plus 0 squared close parentheses times e to the power of negative open parentheses 0 squared plus 0 squared close parentheses end exponent equals 0

      Ponieważ

      H subscript 2 equals H subscript 3 equals 0

      to w punktach P subscript 2 open parentheses 0 comma 1 close parentheses oraz P subscript 3 open parentheses 0 comma negative 1 close parentheses sytuacja jest nieznana (potrzebujemy wiele badań). 

       

      Jednak, jak już mówiono powyżej, współrzędne tych punktów spełniają warunek 

      x squared plus y squared equals 1

      dlatego w tych punktach nie ma ekstrema lokalne.

      Odpowiedź:

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals 0

      5b/4e/4a87bb1fa7976d920fd270009d4b.png” alt=”fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction equals fraction numerator partial differential squared f over denominator partial differential y partial differential x end fraction equals 4″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»x«/mi»«mo»§#8706;«/mo»«mi»y«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»y«/mi»«mo»§#8706;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»4«/mn»«/math»” />

      fraction numerator partial differential squared f over denominator partial differential y squared end fraction equals 12 y squared minus 4

       

      Macierz Hessego ma postać:

      H subscript f left parenthesis x comma y right parenthesis equals open square brackets table row cell 12 x squared minus 4 end cell 4 row 4 cell 12 y squared minus 4 end cell end table close square brackets

       

      H subscript f left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals H subscript f left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals open square brackets table row 20 4 row 4 20 end table close square brackets

      M subscript 1 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 1 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 greater than 0

      M subscript 2 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 2 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 times 20 minus 4 times 4 equals 384 greater than 0

      Zatem w punktach open parentheses square root of 2 comma negative square root of 2 close parentheses oraz open parentheses negative square root of 2 comma square root of 2 close parentheses podana funkcja ma minima lokalne właściwe. 

       

      H subscript f left parenthesis 0 comma 0 right parenthesis equals open square brackets table row cell negative 4 end cell 4 row 4 cell negative 4 end cell end table close square brackets

      M subscript 2 left parenthesis 0 comma 0 right parenthesis equals open parentheses negative 4 close parentheses times open parentheses negative 4 close parentheses minus 4 times 4 equals 0

      Na razie nie wiemy, czy w punkcie open parentheses 0 comma 0 close parentheses funkcja f ma ekstremum. 

      Dla x equals 0 mamy: f left parenthesis 0 comma y right parenthesis equals y to the power of 4 minus 2 y squared. Wtedy punkt open parentheses 0 comma 0 close parentheses to maksimum lokalne funkcji f.

      Dla y equals x mamy:
       f left parenthesis x comma x right parenthesis equals x to the power of 4 plus x to the power of 4 minus 2 x squared plus 4 x squared minus 2 x squared equals 2 x to the power of 4. Wtedy
      punkt open parentheses 0 comma 0 close parentheses to minimum lokalne.

      Zatem w punkcie open parentheses 0 comma 0 close parentheses funkcja f nie posiada ekstremum.

       

      Musimy zbadać jeszcze wartości na brzegach wskazanego obszaru ograniczonego prostymi: x equals 0 comma space y equals 0 comma space x plus y equals 5.

       

  6. Patrycja pisze:

    Czy umie ktoś policzyć tę całkę

    Proszę o pomoc

  7. Angelika pisze:

    Jak obliczyć całkę? (x+9)/(x^3+2x^2+3x)

     

    Proszę o pomoc.

     

    1. Krystian Karczyński pisze:

      Dokładną metodę liczenia całek wymiernych pokazałem tutaj:

      https://online.etrapez.pl/wybor-kursu/calki-nieoznaczone/lekcja-5-calki-wymierne/

      Zastosujmy ją do tego konkretnego przykładu:

      integral fraction numerator x plus 9 over denominator x cubed plus 2 x squared plus 3 x end fraction d x

      W mianowniku mamy wyższy stopień wielomianu niż w liczniku. Mianownik można rozłożyć na czynniki, bo jest stopnia nieparzystego (3). Zgodnie więc z rozpiską:

      integral fraction numerator x plus 9 over denominator x cubed plus 2 x squared plus 3 x end fraction d x equals integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x

      Dalej nie da się rozłożyć, bo delta trójmianu kwadratowego jest ujemna:

      integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x equals
increment equals 2 squared minus 4 times 1 times 3 equals 4 minus 12 equals negative 8

      Ułamek rozkładamy na ułamki proste:

      fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction equals A over x plus fraction numerator B x plus C over denominator x squared plus 2 x plus 3 end fraction space space divided by times x open parentheses x squared plus 2 x plus 3 close parentheses
x plus 9 identical to A open parentheses x squared plus 2 x plus 3 close parentheses plus open parentheses B x plus C close parentheses x
x plus 9 identical to A x squared plus 2 A x plus 3 A plus B x squared plus C x
open curly brackets table row cell 0 equals A plus B end cell row cell 1 equals 2 A plus C end cell row cell 9 equals 3 A end cell end table close
9 equals 3 A
A equals 3
open curly brackets table attributes columnalign left end attributes row cell 0 equals 3 plus B end cell row cell 1 equals 2 times 3 plus C end cell end table close
open curly brackets table attributes columnalign left end attributes row cell B equals negative 3 end cell row cell C equals negative 5 end cell end table close
fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction equals 3 over x plus fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction

      Czyli:

      integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x equals integral 3 over x d x plus integral fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction d x

      Cdn. :)99/38/58957e8897e9f4e5b309cd70cc28.png” alt=”for all for M element of straight real numbers of there exists for N of for all for n greater than N of a subscript n less than M” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo»§#8704;«/mo»«mrow»«mi»M«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«/mrow»«/munder»«munder»«mo»§#8707;«/mo»«mi»N«/mi»«/munder»«munder»«mo»§#8704;«/mo»«mrow»«mi»n«/mi»«mo»§#62;«/mo»«mi»N«/mi»«/mrow»«/munder»«msub»«mi»a«/mi»«mi»n«/mi»«/msub»«mo»§#60;«/mo»«mi»M«/mi»«/math»” />
      Trzeba więc wyjść z nierówności

      a subscript n greater than M

      lub:

      a subscript n less than M

      Ma Pan jakiś konkretny przykład?51/4a/9d54b58ea3e7d8fe464d9a5e5151.png” alt=”f open parentheses x comma y close parentheses equals open parentheses x squared plus y squared close parentheses times e to the power of negative open parentheses x squared plus y squared close parentheses end exponent” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mrow»«/msup»«/math»” /> jest parzysta, to

      A subscript 3 equals A subscript 2 equals 0

      B subscript 3 equals B subscript 2 equals 0

      C subscript 3 equals C subscript 2 equals negative 4 over e

      Dalej liczymy hesjan:

      H equals open vertical bar table row A B row B C end table close vertical bar equals A times C minus B squared

      H subscript 1 equals A subscript 1 times C subscript 1 minus B subscript 1 superscript 2 equals 2 times 2 minus 0 squared equals 4

      H subscript 2 equals A subscript 2 times C subscript 2 minus B subscript 2 superscript 2 equals 0 times open parentheses negative 4 over e close parentheses minus 0 squared equals 0

      H subscript 3 equals H subscript 2 equals 0

      Ponieważ

      H subscript 1 equals 4 greater than 0 comma space A subscript 1 equals 2 greater than 0,

      to w punkcie P subscript 1 open parentheses 0 comma 0 close parentheses funkcja osiąga minimum lokalne, i

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals open parentheses 0 squared plus 0 squared close parentheses times e to the power of negative open parentheses 0 squared plus 0 squared close parentheses end exponent equals 0

      Ponieważ

      H subscript 2 equals H subscript 3 equals 0

      to w punktach P subscript 2 open parentheses 0 comma 1 close parentheses oraz P subscript 3 open parentheses 0 comma negative 1 close parentheses sytuacja jest nieznana (potrzebujemy wiele badań). 

       

      Jednak, jak już mówiono powyżej, współrzędne tych punktów spełniają warunek 

      x squared plus y squared equals 1

      dlatego w tych punktach nie ma ekstrema lokalne.

      Odpowiedź:

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals 0

      5b/4e/4a87bb1fa7976d920fd270009d4b.png” alt=”fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction equals fraction numerator partial differential squared f over denominator partial differential y partial differential x end fraction equals 4″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»x«/mi»«mo»§#8706;«/mo»«mi»y«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»y«/mi»«mo»§#8706;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»4«/mn»«/math»” />

      fraction numerator partial differential squared f over denominator partial differential y squared end fraction equals 12 y squared minus 4

       

      Macierz Hessego ma postać:

      H subscript f left parenthesis x comma y right parenthesis equals open square brackets table row cell 12 x squared minus 4 end cell 4 row 4 cell 12 y squared minus 4 end cell end table close square brackets

       

      H subscript f left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals H subscript f left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals open square brackets table row 20 4 row 4 20 end table close square brackets

      M subscript 1 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 1 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 greater than 0

      M subscript 2 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 2 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 times 20 minus 4 times 4 equals 384 greater than 0

      Zatem w punktach open parentheses square root of 2 comma negative square root of 2 close parentheses oraz open parentheses negative square root of 2 comma square root of 2 close parentheses podana funkcja ma minima lokalne właściwe. 

       

      H subscript f left parenthesis 0 comma 0 right parenthesis equals open square brackets table row cell negative 4 end cell 4 row 4 cell negative 4 end cell end table close square brackets

      M subscript 2 left parenthesis 0 comma 0 right parenthesis equals open parentheses negative 4 close parentheses times open parentheses negative 4 close parentheses minus 4 times 4 equals 0

      Na razie nie wiemy, czy w punkcie open parentheses 0 comma 0 close parentheses funkcja f ma ekstremum. 

      Dla x equals 0 mamy: f left parenthesis 0 comma y right parenthesis equals y to the power of 4 minus 2 y squared. Wtedy punkt open parentheses 0 comma 0 close parentheses to maksimum lokalne funkcji f.

      Dla y equals x mamy:
       f left parenthesis x comma x right parenthesis equals x to the power of 4 plus x to the power of 4 minus 2 x squared plus 4 x squared minus 2 x squared equals 2 x to the power of 4. Wtedy
      punkt open parentheses 0 comma 0 close parentheses to minimum lokalne.

      Zatem w punkcie open parentheses 0 comma 0 close parentheses funkcja f nie posiada ekstremum.

       

      Musimy zbadać jeszcze wartości na brzegach wskazanego obszaru ograniczonego prostymi: x equals 0 comma space y equals 0 comma space x plus y equals 5.

       

    2. Krystian Karczyński pisze:

      cd.

      stack stack integral 3 over x d x with underbrace below with I subscript 1 below plus stack stack integral fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction d x with underbrace below with I subscript 2 below
I subscript 1 equals integral fraction numerator begin display style 3 end style over denominator begin display style x end style end fraction d x equals 3 integral fraction numerator begin display style 1 end style over denominator begin display style x end style end fraction d x equals 3 ln open vertical bar x close vertical bar plus C
I subscript 2 equals integral fraction numerator begin display style negative 3 x minus 5 end style over denominator begin display style x squared plus 2 x plus 3 end style end fraction d x equals integral fraction numerator begin display style negative 3 x minus 5 end style over denominator begin display style x squared plus 2 x plus 3 end style end fraction d x equals integral fraction numerator begin display style negative 3 x minus 5 end style over denominator begin display style 1 open square brackets open parentheses x plus begin inline style fraction numerator 2 over denominator 2 times 1 end fraction end style close parentheses squared minus begin inline style fraction numerator negative 8 over denominator 4 times 1 end fraction end style close square brackets end style end fraction d x equals
increment equals 2 squared minus 4 times 1 times 3 equals 4 minus 12 equals negative 8
equals integral fraction numerator begin display style negative 3 x minus 5 end style over denominator begin display style open parentheses x plus begin inline style 1 end style close parentheses squared plus 2 end style end fraction d x equals integral fraction numerator begin display style negative 3 x end style over denominator begin display style open parentheses x plus begin inline style 1 end style close parentheses squared plus 2 end style end fraction d x plus integral fraction numerator begin display style negative 5 end style over denominator begin display style open parentheses x plus begin inline style 1 end style close parentheses squared plus 2 end style end fraction d x equals
equals negative 3 integral fraction numerator begin display style x end style over denominator begin display style open parentheses x plus begin inline style 1 end style close parentheses squared plus 2 end style end fraction d x minus 5 integral fraction numerator begin display style 1 end style over denominator begin display style open parentheses x plus begin inline style 1 end style close parentheses squared plus 2 end style end fraction d x equals open vertical bar table row cell t equals x plus 1 rightwards double arrow x equals t minus 1 end cell row cell d t equals d x end cell end table close vertical bar equals
equals negative 3 integral fraction numerator begin display style t minus 1 end style over denominator begin display style t squared plus 2 end style end fraction d t minus 5 integral fraction numerator begin display style 1 end style over denominator begin display style t squared plus 2 end style end fraction d t equals negative 3 integral fraction numerator begin display style t end style over denominator begin display style t squared plus 2 end style end fraction d t plus 3 integral fraction numerator begin display style 1 end style over denominator begin display style t squared plus 2 end style end fraction d t minus 5 integral fraction numerator begin display style 1 end style over denominator begin display style t squared plus 2 end style end fraction d t equals
equals negative 3 integral fraction numerator begin display style t end style over denominator begin display style t squared plus 2 end style end fraction d t minus 2 integral fraction numerator begin display style 1 end style over denominator begin display style t squared plus 2 end style end fraction d t equals open vertical bar table row cell u equals t squared plus 2 end cell row cell d u equals 2 t d t end cell row cell t d t equals begin inline style 1 half end style d u end cell end table close vertical bar equals negative 3 integral fraction numerator 1 half begin display style d end style begin display style u end style over denominator u end fraction minus 2 times fraction numerator 1 over denominator square root of 2 end fraction a r c t g fraction numerator t over denominator square root of 2 end fraction plus C equals
equals negative 3 over 2 ln open vertical bar u close vertical bar minus fraction numerator 2 over denominator square root of 2 end fraction a r c t g fraction numerator x plus 1 over denominator square root of 2 end fraction plus C equals negative fraction numerator begin display style 3 end style over denominator begin display style 2 end style end fraction ln open vertical bar t squared plus 2 close vertical bar minus fraction numerator begin display style 2 end style over denominator begin display style square root of 2 end style end fraction a r c t g fraction numerator begin display style x plus 1 end style over denominator begin display style square root of 2 end style end fraction plus C equals
equals negative fraction numerator begin display style 3 end style over denominator begin display style 2 end style end fraction ln open vertical bar open parentheses x plus 1 close parentheses squared plus 2 close vertical bar minus fraction numerator begin display style 2 end style over denominator begin display style square root of 2 end style end fraction a r c t g fraction numerator begin display style x plus 1 end style over denominator begin display style square root of 2 end style end fraction plus C
stack stack integral fraction numerator begin display style 3 end style over denominator begin display style x end style end fraction d x with underbrace below with I subscript 1 below plus stack stack integral fraction numerator begin display style negative 3 x minus 5 end style over denominator begin display style x squared plus 2 x plus 3 end style end fraction d x with underbrace below with I subscript 2 below equals 3 ln open vertical bar x close vertical bar minus fraction numerator begin display style 3 end style over denominator begin display style 2 end style end fraction ln open vertical bar open parentheses x plus 1 close parentheses squared plus 2 close vertical bar minus fraction numerator begin display style 2 end style over denominator begin display style square root of 2 end style end fraction a r c t g fraction numerator begin display style x plus 1 end style over denominator begin display style square root of 2 end style end fraction plus C
a1/71/34f3d48205f417ca8f78d7c0342f.png” alt=”integral fraction numerator x plus 9 over denominator x cubed plus 2 x squared plus 3 x end fraction d x equals integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mfrac»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/mrow»«mrow»«mi»x«/mi»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mi»d«/mi»«mi»x«/mi»«/math»” />

      Dalej nie da się rozłożyć, bo delta trójmianu kwadratowego jest ujemna:

      integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x equals
increment equals 2 squared minus 4 times 1 times 3 equals 4 minus 12 equals negative 8

      Ułamek rozkładamy na ułamki proste:

      fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction equals A over x plus fraction numerator B x plus C over denominator x squared plus 2 x plus 3 end fraction space space divided by times x open parentheses x squared plus 2 x plus 3 close parentheses
x plus 9 identical to A open parentheses x squared plus 2 x plus 3 close parentheses plus open parentheses B x plus C close parentheses x
x plus 9 identical to A x squared plus 2 A x plus 3 A plus B x squared plus C x
open curly brackets table row cell 0 equals A plus B end cell row cell 1 equals 2 A plus C end cell row cell 9 equals 3 A end cell end table close
9 equals 3 A
A equals 3
open curly brackets table attributes columnalign left end attributes row cell 0 equals 3 plus B end cell row cell 1 equals 2 times 3 plus C end cell end table close
open curly brackets table attributes columnalign left end attributes row cell B equals negative 3 end cell row cell C equals negative 5 end cell end table close
fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction equals 3 over x plus fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction

      Czyli:

      integral fraction numerator x plus 9 over denominator x open parentheses x squared plus 2 x plus 3 close parentheses end fraction d x equals integral 3 over x d x plus integral fraction numerator negative 3 x minus 5 over denominator x squared plus 2 x plus 3 end fraction d x

      Cdn. :)99/38/58957e8897e9f4e5b309cd70cc28.png” alt=”for all for M element of straight real numbers of there exists for N of for all for n greater than N of a subscript n less than M” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«munder»«mo»§#8704;«/mo»«mrow»«mi»M«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«/mrow»«/munder»«munder»«mo»§#8707;«/mo»«mi»N«/mi»«/munder»«munder»«mo»§#8704;«/mo»«mrow»«mi»n«/mi»«mo»§#62;«/mo»«mi»N«/mi»«/mrow»«/munder»«msub»«mi»a«/mi»«mi»n«/mi»«/msub»«mo»§#60;«/mo»«mi»M«/mi»«/math»” />
      Trzeba więc wyjść z nierówności

      a subscript n greater than M

      lub:

      a subscript n less than M

      Ma Pan jakiś konkretny przykład?51/4a/9d54b58ea3e7d8fe464d9a5e5151.png” alt=”f open parentheses x comma y close parentheses equals open parentheses x squared plus y squared close parentheses times e to the power of negative open parentheses x squared plus y squared close parentheses end exponent” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mi»y«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mrow»«/msup»«/math»” /> jest parzysta, to

      A subscript 3 equals A subscript 2 equals 0

      B subscript 3 equals B subscript 2 equals 0

      C subscript 3 equals C subscript 2 equals negative 4 over e

      Dalej liczymy hesjan:

      H equals open vertical bar table row A B row B C end table close vertical bar equals A times C minus B squared

      H subscript 1 equals A subscript 1 times C subscript 1 minus B subscript 1 superscript 2 equals 2 times 2 minus 0 squared equals 4

      H subscript 2 equals A subscript 2 times C subscript 2 minus B subscript 2 superscript 2 equals 0 times open parentheses negative 4 over e close parentheses minus 0 squared equals 0

      H subscript 3 equals H subscript 2 equals 0

      Ponieważ

      H subscript 1 equals 4 greater than 0 comma space A subscript 1 equals 2 greater than 0,

      to w punkcie P subscript 1 open parentheses 0 comma 0 close parentheses funkcja osiąga minimum lokalne, i

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals open parentheses 0 squared plus 0 squared close parentheses times e to the power of negative open parentheses 0 squared plus 0 squared close parentheses end exponent equals 0

      Ponieważ

      H subscript 2 equals H subscript 3 equals 0

      to w punktach P subscript 2 open parentheses 0 comma 1 close parentheses oraz P subscript 3 open parentheses 0 comma negative 1 close parentheses sytuacja jest nieznana (potrzebujemy wiele badań). 

       

      Jednak, jak już mówiono powyżej, współrzędne tych punktów spełniają warunek 

      x squared plus y squared equals 1

      dlatego w tych punktach nie ma ekstrema lokalne.

      Odpowiedź:

      f subscript m i n end subscript equals f open parentheses 0 comma 0 close parentheses equals 0

      5b/4e/4a87bb1fa7976d920fd270009d4b.png” alt=”fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction equals fraction numerator partial differential squared f over denominator partial differential y partial differential x end fraction equals 4″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»x«/mi»«mo»§#8706;«/mo»«mi»y«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msup»«mo»§#8706;«/mo»«mn»2«/mn»«/msup»«mi»f«/mi»«/mrow»«mrow»«mo»§#8706;«/mo»«mi»y«/mi»«mo»§#8706;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»4«/mn»«/math»” />

      fraction numerator partial differential squared f over denominator partial differential y squared end fraction equals 12 y squared minus 4

       

      Macierz Hessego ma postać:

      H subscript f left parenthesis x comma y right parenthesis equals open square brackets table row cell 12 x squared minus 4 end cell 4 row 4 cell 12 y squared minus 4 end cell end table close square brackets

       

      H subscript f left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals H subscript f left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals open square brackets table row 20 4 row 4 20 end table close square brackets

      M subscript 1 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 1 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 greater than 0

      M subscript 2 left parenthesis square root of 2 comma negative square root of 2 right parenthesis equals M subscript 2 left parenthesis negative square root of 2 comma square root of 2 right parenthesis equals 20 times 20 minus 4 times 4 equals 384 greater than 0

      Zatem w punktach open parentheses square root of 2 comma negative square root of 2 close parentheses oraz open parentheses negative square root of 2 comma square root of 2 close parentheses podana funkcja ma minima lokalne właściwe. 

       

      H subscript f left parenthesis 0 comma 0 right parenthesis equals open square brackets table row cell negative 4 end cell 4 row 4 cell negative 4 end cell end table close square brackets

      M subscript 2 left parenthesis 0 comma 0 right parenthesis equals open parentheses negative 4 close parentheses times open parentheses negative 4 close parentheses minus 4 times 4 equals 0

      Na razie nie wiemy, czy w punkcie open parentheses 0 comma 0 close parentheses funkcja f ma ekstremum. 

      Dla x equals 0 mamy: f left parenthesis 0 comma y right parenthesis equals y to the power of 4 minus 2 y squared. Wtedy punkt open parentheses 0 comma 0 close parentheses to maksimum lokalne funkcji f.

      Dla y equals x mamy:
       f left parenthesis x comma x right parenthesis equals x to the power of 4 plus x to the power of 4 minus 2 x squared plus 4 x squared minus 2 x squared equals 2 x to the power of 4. Wtedy
      punkt open parentheses 0 comma 0 close parentheses to minimum lokalne.

      Zatem w punkcie open parentheses 0 comma 0 close parentheses funkcja f nie posiada ekstremum.

       

      Musimy zbadać jeszcze wartości na brzegach wskazanego obszaru ograniczonego prostymi: x equals 0 comma space y equals 0 comma space x plus y equals 5.

       

  8. Mariusz pisze:

    Dzień dobry.

    Czy mógłby ktoś policzyć taką całkę:

     

    Z góry dziękuję.

    Pozdrawiam Mariusz

  9. Aga pisze:

    Dzień Dobry

    Czy ktoś zna rozwiązanie poniższej całki nieoznaczonej

    sinx/(1+3x)

  10. Natalia7 pisze:

    Dzień dobry. Czy mógłby ktoś pomóc mi z tą całką? Mam ją całkować przez części, ale nie wychodzi mi i nie wiem jak obliczyć:
    Całka x^2e^x sinxdx

  11. Noga matematyczna pisze:

    Witam nie mogę dać sobie rady z całka
    (Cos2x/√x^7)dx
    Proszę o pomoc
    Pozdrawiam

  12. matematyk pisze:

    ((s^2)+6,5*(10^4)*s+0,5*(10^8)-0,5*(10^4)*s)((s^5)+6,5*(10^4)+(s^4)+0,5*(10^8)*(s^3))

     

    czemu nie mogę tego policzyć???

  13. Paweł pisze:

    Dzień dobry , pomógł by ktoś z taką całeczką, z góry dziękujeintegral subscript blank fraction numerator x over denominator square root of 2 minus 6 x minus 9 x squared right parenthesis end root end fraction d xe4/a7/ff5859ad2093bd1352ef09eafb6a.png” alt=”open parentheses a minus b close parentheses squared equals a squared minus 2 a b plus b squared” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mfenced»«mrow»«mi»a«/mi»«mo»-«/mo»«mi»b«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»a«/mi»«mi»b«/mi»«mo»+«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/math»” />

    integral open parentheses 1 half minus sin x close parentheses squared d x equals integral open parentheses open parentheses 1 half close parentheses squared minus 2 times 1 half times sin x plus open parentheses sin x close parentheses squared close parentheses d x equals integral open parentheses 1 fourth minus sin x plus sin squared x close parentheses d x equals integral 1 fourth d x minus integral sin x space d x plus integral sin squared x space d x equals.... space space

    Pierwsze dwie całki są proste:

    integral 1 fourth d x equals 1 fourth integral 1 d x space equals space bold 1 over bold 4 bold times bold italic x bold plus bold italic C

    integral sin x space d x equals bold minus bold cos bold italic x bold plus bold italic C

    Zostaje trzecia całka z integral sin squared x space d x. Rozwiązanie tej całki z wytłumaczeniem krok po kroku można odnaleźć w lekcji 7 Kursu Całki nieoznaczone:

    https://online.etrapez.pl/wybor-kursu/calki-nieoznaczone/lekcja-7-calki-trygonometryczne/

    Należy skorzystać tu z przekształcenia dodatkowej zależności funkcji trygonometrycznych, mianowicie

    cos 2 x equals cos squared x minus sin squared x . A że chce mieć same sinusy w kwadracie, korzystam jeszcze z jedynki trygonometrycznej

    sin squared x plus cos squared x equals 1 space space rightwards double arrow space cos squared x equals 1 minus sin squared x space space. Stąd:

    cos 2 x equals cos squared x minus sin squared x
cos 2 x equals 1 minus sin squared x minus sin squared x
cos 2 x equals 1 minus 2 sin squared x
2 sin squared x equals 1 minus cos 2 x space space space divided by space colon 2
sin squared x equals 1 half open parentheses 1 minus cos 2 x close parentheses

    Podstawiam to teraz do całki z sinusem kwadrat

    integral sin squared x space d x equals integral open parentheses 1 half open parentheses 1 minus cos 2 x close parentheses close parentheses d x space equals 1 half space integral 1 space d x minus 1 half integral cos 2 x d x equals 1 half times x minus 1 half times 1 half s i n 2 x plus C equals bold 1 over bold 2 bold italic x bold minus bold 1 over bold 4 bold italic s bold italic i bold italic n bold 2 bold italic x bold plus bold italic C

    Wykorzystano tu bezpośredni wzór z dodatkowych wzorów (lub tez można było to obliczyć szybką całką przez podstawienieintegral cos open parentheses a x close parentheses d x equals 1 over a sin open parentheses a x close parentheses plus C

    Ostatnie wychodzi:

    integral open parentheses 1 half minus sin x close parentheses squared d x equals integral open parentheses open parentheses 1 half close parentheses squared minus 2 times 1 half times sin x plus open parentheses sin x close parentheses squared close parentheses d x equals integral open parentheses 1 fourth minus sin x plus sin squared x close parentheses d x equals integral 1 fourth d x minus integral sin x space d x plus integral sin squared x space d x equals

    equals 1 fourth x minus open parentheses negative cos x close parentheses plus 1 half x minus 1 fourth sin 2 x plus C equals 1 fourth x plus cos x plus 2 over 4 x minus 1 fourth sin 2 x plus C equals bold 3 over bold 4 bold italic x bold plus bold cos bold italic x bold minus bold 1 over bold 4 bold sin bold 2 bold italic x bold plus bold italic C bold equals2f/95/03910a63a4c06c86182f2e96e16b.png” alt=”3 A equals 3 space space space divided by colon 3
    bold italic A bold equals bold 1″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mi»A«/mi»«mo»=«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»/«/mo»«mo»:«/mo»«mn»3«/mn»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold-italic¨»A«/mi»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»1«/mn»«/math»” />,             negative 7 comma 5 times 1 plus 2 B equals negative 1
2 B equals negative 1 plus 7 comma 5
2 B equals 6 comma 5 space space space divided by colon 2
B equals 13 over 2 times 1 half equals bold 13 over bold 4,          4 times 1 minus 4 comma 5 times 13 over 4 plus C equals 2
4 minus 9 over 2 times 13 over 4 plus C equals 2
C equals 2 minus 4 plus 117 over 8 equals 117 over 8 minus 16 over 8 equals bold 101 over bold 8    ,    2 times 13 over 4 minus 3 over 2 times 101 over 8 plus lambda equals negative 4
lambda equals negative 4 minus 13 over 2 plus 303 over 16 equals bold 135 over bold 16

    Tak więc na obecną chwilę mamy rozwiązanie postaci:

    integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses x squared plus 13 over 4 x plus 101 over 8 close parentheses times square root of x squared minus 3 x plus 2 end root space plus space 135 over 16 times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction

    Zostaje więc do policzenia tylko całka z pierwiastkiem integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction .

    CDN.

     

    1. Dzień dobry

      Jedziemy zgodnie ze schematem pokazanym w tej Lekcji.

      integral fraction numerator x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x

      Liczymy pochodną tego pod pierwiastkiem w mianowniku:

      M apostrophe equals open parentheses 2 minus 6 x minus 9 x squared close parentheses apostrophe equals negative 6 minus 18 x

      Doprowadzamy część składników licznika do tej pochodnej:

      integral fraction numerator x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals integral fraction numerator open parentheses negative begin display style 1 over 18 end style close parentheses times open parentheses negative 18 close parentheses x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals negative 1 over 18 integral fraction numerator negative 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals
equals negative 1 over 18 integral fraction numerator negative 6 minus 18 x plus 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals horizontal ellipsis

      Rozbijamy całkę na dwie:

      horizontal ellipsis equals negative 1 over 18 integral fraction numerator negative 6 minus 18 x plus 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals negative 1 over 18 open parentheses stack stack integral fraction numerator negative 6 minus 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 1 below plus stack stack integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 2 below close parentheses equals midline horizontal ellipsis

      Obie te całki liczymy osobno:

      I subscript 1 equals integral fraction numerator negative 6 minus 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals open vertical bar table row cell t equals 2 minus 6 x minus 9 x squared end cell row cell d t equals open parentheses negative 6 minus 18 x close parentheses d x end cell end table close vertical bar equals integral fraction numerator 1 over denominator square root of t end fraction d t equals integral t to the power of negative 1 half end exponent d t equals
equals fraction numerator 1 over denominator negative begin display style 1 half end style plus 1 end fraction t to the power of negative 1 half plus 1 end exponent plus C equals fraction numerator 1 over denominator begin display style 1 half end style end fraction t to the power of 1 half end exponent plus C equals 2 square root of t plus C equals 2 square root of 2 minus 6 x minus 9 x squared end root plus C

      I subscript 2 equals integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x

      Sprowadzamy dwumian w mianowniku do postaci a x squared plus b x plus c equals a open square brackets open parentheses x plus fraction numerator b over denominator 2 a end fraction close parentheses squared minus fraction numerator triangle over denominator 4 a squared end fraction close square brackets.

      triangle equals open parentheses negative 6 close parentheses squared minus 4 times open parentheses negative 9 close parentheses times 2 equals 36 plus 72 equals 108

      I subscript 2 equals integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals integral fraction numerator 6 over denominator square root of negative 9 open square brackets open parentheses x plus begin display style fraction numerator negative 6 over denominator 2 times open parentheses negative 9 close parentheses end fraction end style close parentheses squared minus begin display style fraction numerator 108 over denominator 4 times open parentheses negative 9 close parentheses squared end fraction end style close square brackets end root end fraction d x equals
equals integral fraction numerator 6 over denominator square root of negative 9 open square brackets open parentheses x plus begin display style 1 third end style close parentheses squared minus begin display style 1 third end style close square brackets end root end fraction d x equals integral fraction numerator 6 over denominator square root of negative 9 open parentheses x plus 1 third close parentheses squared plus 3 end root end fraction d x equals open vertical bar table row cell t equals x plus 1 third end cell row cell d t equals d x end cell end table close vertical bar equals
equals integral fraction numerator 6 over denominator square root of negative 9 t squared plus 3 end root end fraction d t equals integral fraction numerator 6 over denominator square root of 3 minus 9 t squared end root end fraction d t equals integral fraction numerator 6 over denominator square root of 9 open parentheses begin display style 3 over 9 end style minus t squared close parentheses end root end fraction d t equals integral fraction numerator 6 over denominator square root of 9 square root of 1 third minus t squared end root end fraction d t equals
equals integral fraction numerator 6 over denominator 3 square root of open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses squared minus t squared end root end fraction d t equals 2 integral fraction numerator 1 over denominator square root of open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses squared minus t squared end root end fraction d t equals 2 a r c sin fraction numerator t over denominator fraction numerator 1 over denominator square root of 3 end fraction end fraction plus C equals
equals 2 a r c sin fraction numerator square root of 3 t over denominator 2 end fraction plus C equals 2 a r c sin open square brackets square root of 3 open parentheses x plus 1 third close parentheses close square brackets plus C

      Podstawiamy I subscript 1 comma I subscript 2 tam, gdzie urwaliśmy, i mamy wynik:

      horizontal ellipsis equals negative 1 over 18 open parentheses stack stack integral fraction numerator negative 6 minus 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 1 below plus stack stack integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 2 below close parentheses equals
equals negative 1 over 18 open parentheses 2 square root of 2 minus 6 x minus 9 x squared end root plus 2 a r c sin open square brackets square root of 3 open parentheses x plus 1 third close parentheses close square brackets close parentheses plus C equals
equals negative 1 over 9 square root of 2 minus 6 x minus 9 x squared end root minus 1 over 9 a r c sin open square brackets square root of 3 open parentheses x plus 1 third close parentheses close square brackets plus C equals

       

  14. Paula pisze:

    Witam mam problem z calka 1/cosx  mogłbys mi ja rozpisac? zgory dziekuje i pozdrawiam

  15. Joanna Wojtowicz pisze:

    Jak obliczyc calke (1 na gorze 0 na dole) adalej (3x^3-x^2+2x-4)dx/((x^2-3x+2)^0,5)gdyby sie udalo krok po kroku zebym mogla rozumiec logikepozdrawiamJoanna

    1. Aby rozwiązać całkę integral subscript 0 superscript 1 fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x posłużę się schematem opisanym w Lekcji 6 Kursu Całki Nieoznaczone.  Zacznę oczywiście od całki nieoznaczonej, na koniec podstawię granice.

      Jest to całka typu integral fraction numerator W subscript n greater or equal than 2 end subscript left parenthesis x right parenthesis space d x over denominator square root of a x squared plus b x plus c end root end fraction. Wynik tego typu całki będzie wyglądał następująco:

      integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses A x squared plus B x plus C close parentheses times square root of x squared minus 3 x plus 2 end root space plus space lambda times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction

      Musze teraz znaleźć niewiadome A, B , C oraz lambdę. No i oczywiście rozwiązać całkę z pierwiastkiem. Na powyższe równanie nakładam obustronnie pochodną. Przy obliczaniu pochodnej z wyrażenia open parentheses A x squared plus B x plus C close parentheses times square root of x squared minus 3 x plus 2 end root pamiętam, że  jest to mnożenie dwóch funkcji, więc jadę z wzorem: open parentheses a times b close parentheses apostrophe equals a apostrophe times b plus a times b apostrophe

      integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses A x squared plus B x plus C close parentheses times square root of x squared minus 3 x plus 2 end root space plus space lambda times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction space space space space divided by space open parentheses blank close parentheses apostrophe

      fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction equals space open parentheses A times 2 x plus B times 1 plus 0 close parentheses times square root of x squared minus 3 x plus 2 end root space plus open parentheses A x squared plus B x plus C close parentheses times fraction numerator 1 over denominator 2 square root of x squared minus 3 x plus 2 end root end fraction times open parentheses x squared minus 3 x plus 2 close parentheses apostrophe plus lambda times fraction numerator 1 over denominator square root of x squared minus 3 x plus 2 end root end fraction

      fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction equals space open parentheses 2 A x plus B close parentheses times square root of x squared minus 3 x plus 2 end root space plus open parentheses A x squared plus B x plus C close parentheses times fraction numerator 1 over denominator 2 square root of x squared minus 3 x plus 2 end root end fraction times open parentheses 2 x minus 3 close parentheses plus lambda times fraction numerator 1 over denominator square root of x squared minus 3 x plus 2 end root end fraction space space space space space space space space space space divided by times square root of x squared minus 3 x plus 2 end root

      3 x cubed minus x squared plus 2 x minus 4 space equals space open parentheses 2 A x plus B close parentheses times open parentheses x squared minus 3 x plus 2 close parentheses space plus 1 half open parentheses A x squared plus B x plus C close parentheses times open parentheses 2 x minus 3 close parentheses plus lambda

      Porządkuję prawą stronę i porównuję potęgi przy odpowiednich x-ach wielomianu.

      3 x cubed minus x squared plus 2 x minus 4 space equals space 2 A x cubed minus 6 A x squared plus 4 A x plus B x squared minus 3 B x plus 2 B space plus 1 half open parentheses 2 A x cubed minus 3 A x squared plus 2 B x squared minus 3 B x plus 2 C x minus 3 C close parentheses plus lambda

      3 x cubed minus x squared plus 2 x minus 4 space equals space 2 A x cubed minus 6 A x squared plus 4 A x plus B x squared minus 3 B x plus 2 B space plus A x cubed minus 3 over 2 A x squared plus B x squared minus 3 over 2 B x plus C x minus 3 over 2 C plus lambda

      3 x cubed minus x squared plus 2 x minus 4 space equals space 3 A x cubed space plus space open parentheses negative 7 1 half A plus 2 B close parentheses x squared space plus space open parentheses 4 A minus 4 1 half B plus C close parentheses x plus open parentheses 2 B minus 3 over 2 C plus lambda close parentheses

      open curly brackets table attributes columnalign left end attributes row cell 3 A equals 3 end cell row cell negative 7 comma 5 A plus 2 B equals negative 1 end cell row cell 4 A minus 4 comma 5 B plus C equals 2 end cell row cell 2 B minus 1 comma 5 C plus lambda equals negative 4 end cell end table close
   No to rozwiązuję po kolei ten układ równań.

      3 A equals 3 space space space divided by colon 3
bold italic A bold equals bold 1,             negative 7 comma 5 times 1 plus 2 B equals negative 1
2 B equals negative 1 plus 7 comma 5
2 B equals 6 comma 5 space space space divided by colon 2
B equals 13 over 2 times 1 half equals bold 13 over bold 4,          4 times 1 minus 4 comma 5 times 13 over 4 plus C equals 2
4 minus 9 over 2 times 13 over 4 plus C equals 2
C equals 2 minus 4 plus 117 over 8 equals 117 over 8 minus 16 over 8 equals bold 101 over bold 8    ,    2 times 13 over 4 minus 3 over 2 times 101 over 8 plus lambda equals negative 4
lambda equals negative 4 minus 13 over 2 plus 303 over 16 equals bold 135 over bold 16

      Tak więc na obecną chwilę mamy rozwiązanie postaci:

      integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses x squared plus 13 over 4 x plus 101 over 8 close parentheses times square root of x squared minus 3 x plus 2 end root space plus space 135 over 16 times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction

      Zostaje więc do policzenia tylko całka z pierwiastkiem integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction .

      CDN.

       

    2. Kończąc przykład, całka integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction jest całką typu integral fraction numerator d x over denominator square root of a x squared plus b x plus c end root end fraction , gdzie a equals 1 space comma space b equals negative 3 space comma space c equals 2.

      Zapisuję wielomian z mianownika w postaci: a x squared plus b x plus c space equals space a space open square brackets open parentheses x plus fraction numerator b over denominator 2 a end fraction close parentheses squared minus fraction numerator increment over denominator 4 a end fraction close square brackets (schemat z całek niewymiernych).

      Liczę: increment equals open parentheses negative 3 close parentheses squared minus 4 times 1 times 2 equals 9 minus 8 equals 1  . Stąd:

      integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction equals integral fraction numerator d x over denominator square root of 1 times open square brackets open parentheses x plus fraction numerator negative 3 over denominator 2 times 1 end fraction close parentheses squared minus fraction numerator 1 over denominator 4 times 1 end fraction close square brackets end root end fraction equals integral fraction numerator d x over denominator square root of open parentheses x minus 3 over 2 close parentheses squared minus 1 fourth end root end fraction equals open vertical bar table row cell t equals x minus 3 over 2 end cell row cell d t space equals space d x end cell end table close vertical bar equals integral fraction numerator d t over denominator square root of t squared minus 1 fourth end root end fraction equals...

      Korzystając z wzoru na całkę: integral fraction numerator d t over denominator square root of x squared plus q end root end fraction equals ln space open vertical bar x plus square root of x squared plus q end root close vertical bar plus C wychodzi ostatecznie

      ... equals ln space open vertical bar t plus square root of t squared minus 1 fourth end root close vertical bar plus C equals ln space open vertical bar x minus 3 over 2 plus square root of open parentheses x minus 3 over 2 close parentheses squared minus 1 fourth end root close vertical bar plus C

      Dlatego ostateczna całka z przykładu jest postaci:

      bold integral fraction numerator bold 3 bold x to the power of bold 3 bold minus bold x to the power of bold 2 bold plus bold 2 bold x bold minus bold 4 over denominator square root of bold x to the power of bold 2 bold minus bold 3 bold x bold plus bold 2 end root end fraction bold d bold italic x bold space bold equals bold space open parentheses bold x to the power of bold 2 bold plus bold 13 over bold 4 bold x bold plus bold 101 over bold 8 close parentheses bold times square root of bold x to the power of bold 2 bold minus bold 3 bold x bold plus bold 2 end root bold space bold plus bold space bold 135 over bold 16 bold times bold italic l bold italic n bold space open vertical bar bold x bold minus bold 3 over bold 2 bold plus square root of open parentheses bold x bold minus bold 3 over bold 2 close parentheses to the power of bold 2 bold minus bold 1 over bold 4 end root close vertical bar bold plus bold italic C

      Dziwić może różniący się wynik otrzymany z kalkulatora, tzn.

      integral fraction numerator 3 straight x cubed minus straight x squared plus 2 straight x minus 4 over denominator square root of straight x squared minus 3 straight x plus 2 end root end fraction d x space equals space open parentheses straight x squared plus 13 over 4 straight x plus 101 over 8 close parentheses times square root of straight x squared minus 3 straight x plus 2 end root space plus space 135 over 16 times ln space open vertical bar negative 2 square root of open parentheses straight x minus 3 over 2 close parentheses squared minus 1 fourth end root minus 2 x plus 3 close vertical bar plus C

      Wydawałoby się, że to wyrażenie w logarytmie jest inne. Nic mylnego. Wolfram to chytra sztuka i zawsze próbuje wynik maksymalnie “uładnić” 🙂 Po prostu dobrał sobie taką stałą i wciągnął ją do logarytmu, aby pozbyć się ułamka, tzn:

      ln space open vertical bar negative 2 square root of open parentheses straight x minus 3 over 2 close parentheses squared minus 1 fourth end root minus 2 x plus 3 close vertical bar equals space ln space open vertical bar bold minus bold 2 open square brackets square root of open parentheses straight x minus 3 over 2 close parentheses squared minus 1 fourth end root plus x minus 3 over 2 close square brackets close vertical bar equals l n open parentheses a times b close parentheses equals l n a plus l n b equals space ln space open vertical bar negative 2 close vertical bar plus space ln space open vertical bar square root of open parentheses straight x minus 3 over 2 close parentheses squared minus 1 fourth end root plus x minus 3 over 2 close vertical bar

      gdzie ln|-2| to zwykła stała, podlegająca pod “C”.

       

      To została do policzenia całka oznaczona.

      integral subscript 0 superscript 1 fraction numerator 3 straight x cubed minus straight x squared plus 2 straight x minus 4 over denominator square root of straight x squared minus 3 straight x plus 2 end root end fraction d x space equals space space right enclose open parentheses straight x squared plus 13 over 4 straight x plus 101 over 8 close parentheses times square root of straight x squared minus 3 straight x plus 2 end root space plus space 135 over 16 times ln space open vertical bar straight x minus 3 over 2 plus square root of open parentheses straight x minus 3 over 2 close parentheses squared minus 1 fourth end root close vertical bar space space end enclose subscript 0 superscript 1 equals
equals space space open parentheses 1 squared plus 13 over 4 times 1 plus 101 over 8 close parentheses times square root of 1 squared minus 3 times 1 plus 2 end root space plus space 135 over 16 times ln space open vertical bar 1 minus 3 over 2 plus square root of open parentheses 1 minus 3 over 2 close parentheses squared minus 1 fourth end root close vertical bar space minus space open curly brackets space open parentheses 0 squared plus 13 over 4 times 0 plus 101 over 8 close parentheses times square root of 0 squared minus 3 times 0 plus 2 end root space plus space 135 over 16 times ln space open vertical bar 0 minus 3 over 2 plus square root of open parentheses 0 minus 3 over 2 close parentheses squared minus 1 fourth end root close vertical bar space close curly brackets equals
equals space space open parentheses 1 plus 13 over 4 plus 101 over 8 close parentheses times square root of 0 space plus space 135 over 16 times ln space open vertical bar negative 1 half plus square root of 0 close vertical bar space minus 101 over 8 times square root of 2 space minus space 135 over 16 times ln space open vertical bar negative 3 over 2 plus square root of 2 close vertical bar space equals
equals space 135 over 16 times ln space open vertical bar negative 1 half close vertical bar space minus fraction numerator 101 square root of 2 over denominator 8 end fraction space minus space 135 over 16 times ln space open vertical bar negative 3 over 2 plus square root of 2 close vertical bar spaced1/0c/21a0148351bcbc2b671388e3c081.png” alt=”open curly brackets table attributes columnalign left end attributes row cell 3 A equals 3 end cell row cell negative 7 comma 5 A plus 2 B equals negative 1 end cell row cell 4 A minus 4 comma 5 B plus C equals 2 end cell row cell 2 B minus 1 comma 5 C plus lambda equals negative 4 end cell end table close
      ” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨{¨ close=¨¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mn»3«/mn»«mi»A«/mi»«mo»=«/mo»«mn»3«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»7«/mn»«mo»,«/mo»«mn»5«/mn»«mi»A«/mi»«mo»+«/mo»«mn»2«/mn»«mi»B«/mi»«mo»=«/mo»«mo»-«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»A«/mi»«mo»-«/mo»«mn»4«/mn»«mo»,«/mo»«mn»5«/mn»«mi»B«/mi»«mo»+«/mo»«mi»C«/mi»«mo»=«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«mi»B«/mi»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mn»5«/mn»«mi»C«/mi»«mo»+«/mo»«mi»§#955;«/mi»«mo»=«/mo»«mo»-«/mo»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«mspace linebreak=¨newline¨/»«/math»” />   No to rozwiązuję po kolei ten układ równań.

      3 A equals 3 space space space divided by colon 3
bold italic A bold equals bold 1,             negative 7 comma 5 times 1 plus 2 B equals negative 1
2 B equals negative 1 plus 7 comma 5
2 B equals 6 comma 5 space space space divided by colon 2
B equals 13 over 2 times 1 half equals bold 13 over bold 4,          4 times 1 minus 4 comma 5 times 13 over 4 plus C equals 2
4 minus 9 over 2 times 13 over 4 plus C equals 2
C equals 2 minus 4 plus 117 over 8 equals 117 over 8 minus 16 over 8 equals bold 101 over bold 8    ,    2 times 13 over 4 minus 3 over 2 times 101 over 8 plus lambda equals negative 4
lambda equals negative 4 minus 13 over 2 plus 303 over 16 equals bold 135 over bold 16

      Tak więc na obecną chwilę mamy rozwiązanie postaci:

      integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses x squared plus 13 over 4 x plus 101 over 8 close parentheses times square root of x squared minus 3 x plus 2 end root space plus space 135 over 16 times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction

      Zostaje więc do policzenia tylko całka z pierwiastkiem integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction .

      CDN.

       

  16. Karol pisze:

    Jak obliczyć y’-y*sinx=sinx*cosx.
    Czy da się to zadanie wpisać w kalkulator?

    1. Da się wpisać w kalkulator, tyle, że w inny, do równań różniczkowych:

      https://blog.etrapez.pl/narzedzia/kalkulatory/kalkulator-do-rownan-rozniczkowych/

      A obliczyć można stosując metodę “uzmienniania stałej”, pokazałem ją w swoim Kursie dokładniej:

      y apostrophe minus y sin x equals sin x cos x

      Rozwiązuję odpowiadające temu równaniu równanie jednorodne:

      y apostrophe minus y sin x equals 0 y apostrophe equals y sin x fraction numerator d y over denominator d x end fraction equals y sin x space space divided by colon y space space divided by times d x fraction numerator d y over denominator y end fraction equals sin x d x integral fraction numerator d y over denominator y end fraction equals integral sin x d x ln open vertical bar y close vertical bar equals negative cos x plus C y equals e to the power of negative cos x plus C end exponent y equals e to the power of negative cos x end exponent e to the power of C y equals C e to the power of negative cos x end exponent

      W otrzymanym rozwiązaniu równania jednorodnego “uzmienniamy stałą”, a następnie liczymy pochodną:

      y equals C open parentheses x close parentheses e to the power of negative cos x end exponent y apostrophe equals open square brackets C open parentheses x close parentheses e to the power of negative cos x end exponent close square brackets apostrophe equals C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent plus C open parentheses x close parentheses open parentheses e to the power of negative cos x end exponent close parentheses apostrophe equals C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent plus C open parentheses x close parentheses e to the power of negative cos x end exponent open parentheses negative cos x close parentheses apostrophe equals C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent plus C open parentheses x close parentheses e to the power of negative cos x end exponent sin x

      Otrzymane wyniki wstawiamy do równania na początku:

      C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent plus C open parentheses x close parentheses e to the power of negative cos x end exponent sin x minus C open parentheses x close parentheses e to the power of negative cos x end exponent sin x equals sin x cos x C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent equals sin x cos x C apostrophe open parentheses x close parentheses 1 over e to the power of cos x end exponent equals sin x cos x space space divided by times e to the power of cos x end exponent C apostrophe open parentheses x close parentheses equals e to the power of cos x end exponent sin x cos x C open parentheses x close parentheses equals integral e to the power of cos x end exponent sin x cos x d x

      Całeczkę rozwiązuję najpierw przez podstawienie, potem przez części:

      integral e to the power of cos x end exponent sin x cos x d x equals open vertical bar table row cell table row cell t equals cos x end cell row cell d t equals open parentheses negative sin x close parentheses d x end cell end table end cell row cell negative d t equals sin x d x end cell row blank end table close vertical bar equals negative integral e to the power of t times t d t equals open vertical bar table row cell u open parentheses t close parentheses equals t end cell cell v apostrophe open parentheses t close parentheses equals e to the power of t end cell row cell u apostrophe open parentheses t close parentheses equals 1 end cell cell v open parentheses t close parentheses equals e to the power of t end cell end table close vertical bar equals equals negative open parentheses t e to the power of t minus integral 1 times e to the power of t d t close parentheses equals negative t e to the power of t plus integral e to the power of t d t equals negative t e to the power of t plus e to the power of t plus C equals negative cos x e to the power of cos x end exponent plus e to the power of cos x end exponent plus C

      Mam więc, że:

      C open parentheses x close parentheses equals negative cos x e to the power of cos x end exponent plus e to the power of cos x end exponent plus C

      Wstawiam ten wynik do rozwiązania z uzmiennioną stałą:

      y equals C open parentheses x close parentheses e to the power of negative cos x end exponent y equals open parentheses negative cos x e to the power of cos x end exponent plus e to the power of cos x end exponent plus C close parentheses e to the power of negative cos x end exponent equals negative cos x plus 1 plus C e to the power of negative cos x end exponent

      Co jest rozwiązaniem tego równania różniczkowego.df/13/e5a61b435e59e4fa320ddb9dd062.png” alt=”negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«msup»«mi»ln«/mi»«mn»2«/mn»«/msup»«mn»2«/mn»«mo»§#183;«/mo»«mo»§#8747;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»d«/mo»«mi»x«/mi»«/math»” />

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  17. Kasia pisze:

    Witam, mam problem z wyznaczeniem obszaru ograniczonego przez:y=|cosx| , x=0 , x=3/2*pi , y=0.Czy ma ktoś może pomysł jak ją rozwiązać ? (wstawiam poprawione bo było nieczytelne)b9/9f/a39b95d82337db4f7e208571f6c5.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 space minus space 4 to the power of x end root end fraction d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mi»d«/mi»«mi»x«/mi»«/math»” /> 3. integral fraction numerator x over denominator square root of 2 plus 2 x squared end root end fraction d x6b/aa/e5ff7f7200b2935730cfc4755204.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«/mfenced»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»ln«/mi»«mn»2«/mn»«mo»§#160;«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” /> 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  18. Kasia pisze:

    Witam, mam problem z wyznaczeniem obszaru ograniczonego przez:y=|cosx|x=0x=3/2piy=0.Czy ma ktoś może pomysł jak ją rozwiązać ?b9/9f/a39b95d82337db4f7e208571f6c5.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 space minus space 4 to the power of x end root end fraction d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mi»d«/mi»«mi»x«/mi»«/math»” /> 3. integral fraction numerator x over denominator square root of 2 plus 2 x squared end root end fraction d x6b/aa/e5ff7f7200b2935730cfc4755204.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«/mfenced»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»ln«/mi»«mn»2«/mn»«mo»§#160;«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” /> 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  19. Sonia pisze:

    Obliczyć całkę z funkcji e^{x^2}. Jak?

  20. Iva pisze:

    Witam
    Mam problem z całka nieoznaczoną
    2x-9/3pirwiastki z 3

    1.  integral fraction numerator 2 x minus 9 over denominator 3 square root of 3 end fraction d x

      Mianownik to zwykła liczba (nie ma tam zmiennej “x”). Dlatego wyciągam go jako stałą przed całkę. Dalej to proste działania na całkach elementarnych.

      integral fraction numerator 2 x minus 9 over denominator 3 square root of 3 end fraction d x equals integral fraction numerator 1 over denominator 3 square root of 3 end fraction times open parentheses 2 x minus 9 close parentheses space d x space equals space fraction numerator 1 over denominator 3 square root of 3 end fraction times integral open parentheses 2 x minus 9 close parentheses space d x equals

      equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets integral 2 x space d x minus integral 9 space d x close square brackets equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets 2 times integral x space d x minus 9 integral 1 space d x close square brackets equals

      equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets 2 times 1 half x squared space minus 9 times x space plus space C close square brackets equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets x squared space minus 9 x space plus space C close square brackets equals fraction numerator bold x to the power of bold 2 bold space bold minus bold 9 bold x over denominator bold 3 square root of bold 3 end fraction bold plus bold space bold italic C

      63/30/de0802a71e365eb67f083ad013c0.png” alt=”integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»t«/mi»«mo»+«/mo»«mi»C«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/math»” />3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

      negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  21. Arek pisze:

    Witam mam problem z niektórymi przykładami całkowania przez podstawienie, mógłby ktoś mi je rozpisać? 1.  integral fraction numerator sin open parentheses square root of x close parentheses over denominator square root of x end fraction d x space,  2. integral fraction numerator 2 to the power of x over denominator square root of 1 space minus space 4 to the power of x end root end fraction d x 3. integral fraction numerator x over denominator square root of 2 plus 2 x squared end root end fraction d x6b/aa/e5ff7f7200b2935730cfc4755204.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«/mfenced»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»ln«/mi»«mn»2«/mn»«mo»§#160;«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” /> 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    1. Całka nr 1:  (pójdzie przez podstawienie)

       integral fraction numerator sin open parentheses square root of x close parentheses over denominator square root of x end fraction d x space equals integral sin open parentheses square root of x close parentheses times fraction numerator 1 over denominator square root of x end fraction d x equals open vertical bar table row cell square root of x equals t end cell row cell fraction numerator 1 over denominator 2 square root of x end fraction space d x space equals d t end cell row cell fraction numerator 1 over denominator square root of x end fraction space d x space equals 2 space d t end cell end table close vertical bar space equals

      equals integral sin open parentheses t close parentheses times 2 space d t equals space 2 integral sin t space d t equals 2 times open parentheses negative cos t close parentheses plus C space equals space bold minus bold 2 bold times bold italic c bold italic o bold italic s open parentheses square root of bold x close parentheses bold plus bold italic C bold space

      Dokładny sposób postępowania omówiono w Kursie Całki Nieoznaczone. Pierwsza lekcja (całki bezpośrednie) dostępna jest za darmo po założeniu konta na Akademii.3d/4d/a1cf1080682b4d0b1547d815ffec.png” alt=”equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets integral 2 x space d x minus integral 9 space d x close square brackets equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets 2 times integral x space d x minus 9 integral 1 space d x close square brackets equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«/mfrac»«mo»§#183;«/mo»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8747;«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#160;«/mo»«mo»d«/mo»«mi»x«/mi»«mo»-«/mo»«mo»§#8747;«/mo»«mn»9«/mn»«mo»§#160;«/mo»«mo»d«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«/mfrac»«mo»§#183;«/mo»«mfenced open=¨[¨ close=¨]¨»«mrow»«mn»2«/mn»«mo»§#183;«/mo»«mo»§#8747;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»d«/mo»«mi»x«/mi»«mo»-«/mo»«mn»9«/mn»«mo»§#8747;«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mo»d«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mo»=«/mo»«/math»” />

      equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets 2 times 1 half x squared space minus 9 times x space plus space C close square brackets equals fraction numerator 1 over denominator 3 square root of 3 end fraction times open square brackets x squared space minus 9 x space plus space C close square brackets equals fraction numerator bold x to the power of bold 2 bold space bold minus bold 9 bold x over denominator bold 3 square root of bold 3 end fraction bold plus bold space bold italic C

      63/30/de0802a71e365eb67f083ad013c0.png” alt=”integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»t«/mi»«mo»+«/mo»«mi»C«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/math»” />3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

      negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    2. Całka nr 2 także pójdzie przez podstawienie i wykorzystanie bezpośredniego wzoru

      integral fraction numerator 1 over denominator square root of 1 space minus space x squared end root end fraction d x equals a r c sin x space plus space C (wzór 15.)

      oraz małe przekształcanie 4 to the power of x equals open parentheses 2 squared close parentheses to the power of x equals 2 to the power of 2 times x end exponent equals open parentheses 2 to the power of x close parentheses squared

       integral fraction numerator 2 to the power of x over denominator square root of 1 space minus space 4 to the power of x end root end fraction d x space equals space open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x times ln 2 space d x space equals space d t end cell row cell 2 to the power of x space d x space equals fraction numerator 1 over denominator ln 2 end fraction space d t end cell end table close vertical bar space equals space integral fraction numerator t over denominator square root of 1 space minus space t squared end root end fraction times space fraction numerator 1 over denominator ln 2 end fraction space d t space equals space

      equals space fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator t over denominator square root of 1 space minus space t squared end root end fraction space d t space equals space fraction numerator 1 over denominator ln 2 end fraction times a r c s i n left parenthesis t right parenthesis space plus C space equals space fraction numerator bold 1 over denominator bold l bold n bold 2 end fraction bold times bold italic a bold italic r bold italic c bold italic s bold italic i bold italic n bold left parenthesis bold 2 to the power of bold x bold right parenthesis bold space bold plus bold italic C63/30/de0802a71e365eb67f083ad013c0.png” alt=”integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mi»t«/mi»«mo»+«/mo»«mi»C«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»sin«/mi»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/math»” />3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

      negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    3. integral fraction numerator x over denominator square root of 2 plus 2 x squared end root end fraction d x ta całka także pójdzie przez podstawienie (całego wyrażenia pod pierwiastkiem):

      integral fraction numerator x over denominator square root of 2 plus 2 x squared end root end fraction d x equals integral fraction numerator 1 over denominator square root of 2 plus 2 x squared end root end fraction times x space d x equals open vertical bar table row cell 2 plus 2 x squared equals t end cell row cell 0 plus 2 times 2 x space d x equals d t end cell row cell x space d x equals 1 fourth d t end cell end table close vertical bar space equals space integral fraction numerator 1 over denominator square root of t end fraction times 1 fourth d t equals

      equals space 1 fourth integral space t to the power of negative 1 half end exponent d t equals 1 fourth times 1 over blank to the power of negative 1 half plus 1 end exponent space t space to the power of negative 1 half plus 1 end exponent plus C space equals space 1 fourth times 1 over blank to the power of 1 half end exponent space t space to the power of 1 half end exponent plus C space equals space 1 fourth times 2 square root of space t end root plus C space equals bold 1 over bold 2 square root of bold space bold 2 bold plus bold 2 bold x to the power of bold 2 end root bold plus bold italic C

      6b/44/bb2c76786f906e50f4373ea08201.png” alt=”C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent plus C open parentheses x close parentheses e to the power of negative cos x end exponent sin x minus C open parentheses x close parentheses e to the power of negative cos x end exponent sin x equals sin x cos x C apostrophe open parentheses x close parentheses e to the power of negative cos x end exponent equals sin x cos x C apostrophe open parentheses x close parentheses 1 over e to the power of cos x end exponent equals sin x cos x space space divided by times e to the power of cos x end exponent C apostrophe open parentheses x close parentheses equals e to the power of cos x end exponent sin x cos x C open parentheses x close parentheses equals integral e to the power of cos x end exponent sin x cos x d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mo»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mi»C«/mi»«mfenced»«mi»x«/mi»«/mfenced»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mi»sin«/mi»«mi»x«/mi»«mo»-«/mo»«mi»C«/mi»«mfenced»«mi»x«/mi»«/mfenced»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mi»sin«/mi»«mi»x«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mspace linebreak=¨newline¨/»«mi»C«/mi»«mo»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«msup»«mi»e«/mi»«mrow»«mo»-«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mspace linebreak=¨newline¨/»«mi»C«/mi»«mo»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mfrac»«mn»1«/mn»«msup»«mi»e«/mi»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«/mfrac»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»/«/mo»«mo»§#183;«/mo»«msup»«mi»e«/mi»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mspace linebreak=¨newline¨/»«mi»C«/mi»«mo»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»e«/mi»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mspace linebreak=¨newline¨/»«mi»C«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8747;«/mo»«msup»«mi»e«/mi»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/msup»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mi»d«/mi»«mi»x«/mi»«/math»” />

      Całeczkę rozwiązuję najpierw przez podstawienie, potem przez części:

      integral e to the power of cos x end exponent sin x cos x d x equals open vertical bar table row cell table row cell t equals cos x end cell row cell d t equals open parentheses negative sin x close parentheses d x end cell end table end cell row cell negative d t equals sin x d x end cell row blank end table close vertical bar equals negative integral e to the power of t times t d t equals open vertical bar table row cell u open parentheses t close parentheses equals t end cell cell v apostrophe open parentheses t close parentheses equals e to the power of t end cell row cell u apostrophe open parentheses t close parentheses equals 1 end cell cell v open parentheses t close parentheses equals e to the power of t end cell end table close vertical bar equals equals negative open parentheses t e to the power of t minus integral 1 times e to the power of t d t close parentheses equals negative t e to the power of t plus integral e to the power of t d t equals negative t e to the power of t plus e to the power of t plus C equals negative cos x e to the power of cos x end exponent plus e to the power of cos x end exponent plus C

      Mam więc, że:

      C open parentheses x close parentheses equals negative cos x e to the power of cos x end exponent plus e to the power of cos x end exponent plus C

      Wstawiam ten wynik do rozwiązania z uzmiennioną stałą:

      y equals C open parentheses x close parentheses e to the power of negative cos x end exponent y equals open parentheses negative cos x e to the power of cos x end exponent plus e to the power of cos x end exponent plus C close parentheses e to the power of negative cos x end exponent equals negative cos x plus 1 plus C e to the power of negative cos x end exponent

      Co jest rozwiązaniem tego równania różniczkowego.df/13/e5a61b435e59e4fa320ddb9dd062.png” alt=”negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«msup»«mi»ln«/mi»«mn»2«/mn»«/msup»«mn»2«/mn»«mo»§#183;«/mo»«mo»§#8747;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»d«/mo»«mi»x«/mi»«/math»” />

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  22. Łukasz pisze:

    (1/2-sinx)^2 pomoże mi ktoś przy tej całce ?

    1. Chodzi o całkę:  integral open parentheses 1 half minus sin x close parentheses squared d x ?  Jeśli tak to rozwiązanie będzie następujące:

      Najpierw korzystam ze wzoru skróconego mnożenia: open parentheses a minus b close parentheses squared equals a squared minus 2 a b plus b squared

      integral open parentheses 1 half minus sin x close parentheses squared d x equals integral open parentheses open parentheses 1 half close parentheses squared minus 2 times 1 half times sin x plus open parentheses sin x close parentheses squared close parentheses d x equals integral open parentheses 1 fourth minus sin x plus sin squared x close parentheses d x equals integral 1 fourth d x minus integral sin x space d x plus integral sin squared x space d x equals.... space space

      Pierwsze dwie całki są proste:

      integral 1 fourth d x equals 1 fourth integral 1 d x space equals space bold 1 over bold 4 bold times bold italic x bold plus bold italic C

      integral sin x space d x equals bold minus bold cos bold italic x bold plus bold italic C

      Zostaje trzecia całka z integral sin squared x space d x. Rozwiązanie tej całki z wytłumaczeniem krok po kroku można odnaleźć w lekcji 7 Kursu Całki nieoznaczone:

      https://online.etrapez.pl/wybor-kursu/calki-nieoznaczone/lekcja-7-calki-trygonometryczne/

      Należy skorzystać tu z przekształcenia dodatkowej zależności funkcji trygonometrycznych, mianowicie

      cos 2 x equals cos squared x minus sin squared x . A że chce mieć same sinusy w kwadracie, korzystam jeszcze z jedynki trygonometrycznej

      sin squared x plus cos squared x equals 1 space space rightwards double arrow space cos squared x equals 1 minus sin squared x space space. Stąd:

      cos 2 x equals cos squared x minus sin squared x
cos 2 x equals 1 minus sin squared x minus sin squared x
cos 2 x equals 1 minus 2 sin squared x
2 sin squared x equals 1 minus cos 2 x space space space divided by space colon 2
sin squared x equals 1 half open parentheses 1 minus cos 2 x close parentheses

      Podstawiam to teraz do całki z sinusem kwadrat

      integral sin squared x space d x equals integral open parentheses 1 half open parentheses 1 minus cos 2 x close parentheses close parentheses d x space equals 1 half space integral 1 space d x minus 1 half integral cos 2 x d x equals 1 half times x minus 1 half times 1 half s i n 2 x plus C equals bold 1 over bold 2 bold italic x bold minus bold 1 over bold 4 bold italic s bold italic i bold italic n bold 2 bold italic x bold plus bold italic C

      Wykorzystano tu bezpośredni wzór z dodatkowych wzorów (lub tez można było to obliczyć szybką całką przez podstawienieintegral cos open parentheses a x close parentheses d x equals 1 over a sin open parentheses a x close parentheses plus C

      Ostatnie wychodzi:

      integral open parentheses 1 half minus sin x close parentheses squared d x equals integral open parentheses open parentheses 1 half close parentheses squared minus 2 times 1 half times sin x plus open parentheses sin x close parentheses squared close parentheses d x equals integral open parentheses 1 fourth minus sin x plus sin squared x close parentheses d x equals integral 1 fourth d x minus integral sin x space d x plus integral sin squared x space d x equals

      equals 1 fourth x minus open parentheses negative cos x close parentheses plus 1 half x minus 1 fourth sin 2 x plus C equals 1 fourth x plus cos x plus 2 over 4 x minus 1 fourth sin 2 x plus C equals bold 3 over bold 4 bold italic x bold plus bold cos bold italic x bold minus bold 1 over bold 4 bold sin bold 2 bold italic x bold plus bold italic C bold equals2f/95/03910a63a4c06c86182f2e96e16b.png” alt=”3 A equals 3 space space space divided by colon 3
      bold italic A bold equals bold 1″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mi»A«/mi»«mo»=«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»/«/mo»«mo»:«/mo»«mn»3«/mn»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold-italic¨»A«/mi»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»1«/mn»«/math»” />,             negative 7 comma 5 times 1 plus 2 B equals negative 1
2 B equals negative 1 plus 7 comma 5
2 B equals 6 comma 5 space space space divided by colon 2
B equals 13 over 2 times 1 half equals bold 13 over bold 4,          4 times 1 minus 4 comma 5 times 13 over 4 plus C equals 2
4 minus 9 over 2 times 13 over 4 plus C equals 2
C equals 2 minus 4 plus 117 over 8 equals 117 over 8 minus 16 over 8 equals bold 101 over bold 8    ,    2 times 13 over 4 minus 3 over 2 times 101 over 8 plus lambda equals negative 4
lambda equals negative 4 minus 13 over 2 plus 303 over 16 equals bold 135 over bold 16

      Tak więc na obecną chwilę mamy rozwiązanie postaci:

      integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses x squared plus 13 over 4 x plus 101 over 8 close parentheses times square root of x squared minus 3 x plus 2 end root space plus space 135 over 16 times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction

      Zostaje więc do policzenia tylko całka z pierwiastkiem integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction .

      CDN.

       

  23. Jagoda pisze:

    Hej :)pomoże ktoś obliczyć:

  24. Sonia pisze:

    Mam problem z taką oto całką: (x-13)/(x^3-x-6) dx    mianownik ni jak da się rozłożyć – zatem zgodnie ze schematem muszę przekształcić licznik tak, aby był podobny do mianownika – tylko co jeśli się nie da jak tutaj?? HELP 🙂

  25. Filipides pisze:

    a co z taką całką? Całka: 2x/(x-1)^8 dx ?

    1. Ta całka pójdzie tak: integral fraction numerator 2 x over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction d x equals 2 times integral fraction numerator x over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction d x space equals space...

      rozbijam ułamek na sumę dwóch ułamków. Na dole jest aż 8 potęga, ja nie potrzebuję rozbijać na osiem kolejnych ułamków, gdzie w mianowniku będą kolejne potęgi wyrażenia (x-1)… Wystarczy że zejdę tylko o JEDEN stopień niżej, ponieważ w liczniku jest tylko “x” (postać liniowa wielomianu, x w pierwszej potędze), czyli:

      fraction numerator x over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction equals fraction numerator A over denominator left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus fraction numerator B over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction space space space space space space space space divided by times left parenthesis x minus 1 right parenthesis to the power of 8

      x space equals space A open parentheses x minus 1 close parentheses space plus space B

      x space equals space A x minus A space plus space B

      open curly brackets table attributes columnalign left end attributes row cell A equals 1 end cell row cell negative A plus B equals 0 end cell end table close space rightwards double arrow space space space space open curly brackets table attributes columnalign left end attributes row cell A equals 1 end cell row cell B equals 1 end cell end table close

      fraction numerator x over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction equals fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction

      ... equals 2 times integral open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction close square brackets d x equals 2 integral fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 7 end fraction d x plus 2 integral fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction d x equals

      Każdą z całek liczę na boku, przez podstawienie:

      integral fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 7 end fraction d x equals open vertical bar table row cell x minus 1 equals t end cell row cell d x equals d t end cell end table close vertical bar equals integral 1 over t to the power of 7 d t equals integral t to the power of negative 7 end exponent d t equals fraction numerator 1 over denominator negative 7 plus 1 end fraction t to the power of negative 7 plus 1 end exponent plus C equals negative 1 over 6 t to the power of negative 6 end exponent plus C equals negative fraction numerator 1 over denominator 6 left parenthesis x minus 1 right parenthesis to the power of 6 end fraction plus C

      integral fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction d x equals open vertical bar table row cell x minus 1 equals s end cell row cell d x equals d s end cell end table close vertical bar equals integral 1 over s to the power of 8 d s equals integral t to the power of negative 8 end exponent d s equals fraction numerator 1 over denominator negative 8 plus 1 end fraction s to the power of negative 8 plus 1 end exponent plus C equals negative 1 over 7 s to the power of negative 7 end exponent plus C equals negative fraction numerator 1 over denominator 7 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus C

      Wracając do całki:

      equals integral fraction numerator 2 over denominator left parenthesis x minus 1 right parenthesis to the power of 7 end fraction d x plus integral fraction numerator 2 over denominator left parenthesis x minus 1 right parenthesis to the power of 8 end fraction d x equals negative fraction numerator 2 over denominator 6 left parenthesis x minus 1 right parenthesis to the power of 6 end fraction minus fraction numerator 2 over denominator 7 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus C equals negative fraction numerator 1 over denominator 3 left parenthesis x minus 1 right parenthesis to the power of 6 end fraction minus fraction numerator 2 over denominator 7 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus C equals

      equals negative fraction numerator 7 left parenthesis x minus 1 right parenthesis over denominator 21 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction minus fraction numerator 6 over denominator 21 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus C equals fraction numerator negative 7 x plus 7 over denominator 21 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus fraction numerator negative 6 over denominator 21 left parenthesis x minus 1 right parenthesis to the power of 7 end fraction plus C equals fraction numerator bold 1 bold minus bold 7 bold x over denominator bold 21 bold left parenthesis bold x bold minus bold 1 bold right parenthesis to the power of bold 7 end fraction bold plus bold italic C33/37/302f774d6427a377f912f859455b.png” alt=”integral open parentheses 1 half minus sin x close parentheses squared d x equals integral open parentheses open parentheses 1 half close parentheses squared minus 2 times 1 half times sin x plus open parentheses sin x close parentheses squared close parentheses d x equals integral open parentheses 1 fourth minus sin x plus sin squared x close parentheses d x equals integral 1 fourth d x minus integral sin x space d x plus integral sin squared x space d x equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«msup»«mfenced»«mrow»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»-«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfenced»«mrow»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mo»§#183;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#183;«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mfenced»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfenced»«mrow»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»-«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«/mfenced»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»-«/mo»«mo»§#8747;«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»§#160;«/mo»«mo»d«/mo»«mi»x«/mi»«mo»+«/mo»«mo»§#8747;«/mo»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#160;«/mo»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«/math»” />

      equals 1 fourth x minus open parentheses negative cos x close parentheses plus 1 half x minus 1 fourth sin 2 x plus C equals 1 fourth x plus cos x plus 2 over 4 x minus 1 fourth sin 2 x plus C equals bold 3 over bold 4 bold italic x bold plus bold cos bold italic x bold minus bold 1 over bold 4 bold sin bold 2 bold italic x bold plus bold italic C bold equals2f/95/03910a63a4c06c86182f2e96e16b.png” alt=”3 A equals 3 space space space divided by colon 3
      bold italic A bold equals bold 1″ align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mi»A«/mi»«mo»=«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»/«/mo»«mo»:«/mo»«mn»3«/mn»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold-italic¨»A«/mi»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»1«/mn»«/math»” />,             negative 7 comma 5 times 1 plus 2 B equals negative 1
2 B equals negative 1 plus 7 comma 5
2 B equals 6 comma 5 space space space divided by colon 2
B equals 13 over 2 times 1 half equals bold 13 over bold 4,          4 times 1 minus 4 comma 5 times 13 over 4 plus C equals 2
4 minus 9 over 2 times 13 over 4 plus C equals 2
C equals 2 minus 4 plus 117 over 8 equals 117 over 8 minus 16 over 8 equals bold 101 over bold 8    ,    2 times 13 over 4 minus 3 over 2 times 101 over 8 plus lambda equals negative 4
lambda equals negative 4 minus 13 over 2 plus 303 over 16 equals bold 135 over bold 16

      Tak więc na obecną chwilę mamy rozwiązanie postaci:

      integral fraction numerator 3 x cubed minus x squared plus 2 x minus 4 over denominator square root of x squared minus 3 x plus 2 end root end fraction d x space equals space open parentheses x squared plus 13 over 4 x plus 101 over 8 close parentheses times square root of x squared minus 3 x plus 2 end root space plus space 135 over 16 times integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction

      Zostaje więc do policzenia tylko całka z pierwiastkiem integral fraction numerator d x over denominator square root of x squared minus 3 x plus 2 end root end fraction .

      CDN.

       

  26. Wera pisze:

    A pomoże ktoś w takiej całeczce: ln(2x+3)/2x+3

    1. Całka integral fraction numerator ln left parenthesis 2 x plus 3 right parenthesis over denominator 2 x plus 3 end fraction pójdzie tak (przez “podwójne” podstawienie):

      integral fraction numerator ln left parenthesis 2 x plus 3 right parenthesis over denominator 2 x plus 3 end fraction d x equals open vertical bar table row cell 2 x plus 3 equals t end cell row cell 2 space d x equals d t end cell row cell d x equals 1 half d t end cell end table close vertical bar space equals space integral fraction numerator ln left parenthesis t right parenthesis over denominator t end fraction times 1 half d t space equals space 1 half integral ln left parenthesis t right parenthesis times 1 over t d t space equals
equals open vertical bar table row cell l n left parenthesis t right parenthesis equals u end cell row cell 1 over t d t equals d u end cell end table close vertical bar space equals space 1 half integral u space d u space equals space 1 half times 1 half u squared plus C space equals space space 1 fourth u squared plus C space equals space
equals space 1 fourth open square brackets l n left parenthesis t right parenthesis close square brackets space squared plus C space equals space space 1 fourth l n squared left parenthesis t right parenthesis space plus C space equals space bold 1 over bold 4 bold italic l bold italic n to the power of bold 2 bold left parenthesis bold 2 bold italic x bold plus bold 3 bold right parenthesis bold space bold plus bold italic C8c/df/23daf2ae6cdb11fa0eb7fb92ca8a.png” alt=”integral fraction numerator x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals integral fraction numerator open parentheses negative begin display style 1 over 18 end style close parentheses times open parentheses negative 18 close parentheses x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals negative 1 over 18 integral fraction numerator negative 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals
      equals negative 1 over 18 integral fraction numerator negative 6 minus 18 x plus 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals horizontal ellipsis” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mi»x«/mi»«msqrt»«mn»2«/mn»«mo»-«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mrow»«mfenced»«mrow»«mo»-«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mn»18«/mn»«/mfrac»«/mstyle»«/mrow»«/mfenced»«mo»§#183;«/mo»«mfenced»«mrow»«mo»-«/mo»«mn»18«/mn»«/mrow»«/mfenced»«mi»x«/mi»«/mrow»«msqrt»«mn»2«/mn»«mo»-«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»18«/mn»«/mfrac»«mo»§#8747;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»18«/mn»«mi»x«/mi»«/mrow»«msqrt»«mn»2«/mn»«mo»-«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»18«/mn»«/mfrac»«mo»§#8747;«/mo»«mfrac»«mrow»«mo»-«/mo»«mn»6«/mn»«mo»-«/mo»«mn»18«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«msqrt»«mn»2«/mn»«mo»-«/mo»«mn»6«/mn»«mi»x«/mi»«mo»-«/mo»«mn»9«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8230;«/mo»«/math»” />

      Rozbijamy całkę na dwie:

      horizontal ellipsis equals negative 1 over 18 integral fraction numerator negative 6 minus 18 x plus 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals negative 1 over 18 open parentheses stack stack integral fraction numerator negative 6 minus 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 1 below plus stack stack integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 2 below close parentheses equals midline horizontal ellipsis

      Obie te całki liczymy osobno:

      I subscript 1 equals integral fraction numerator negative 6 minus 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals open vertical bar table row cell t equals 2 minus 6 x minus 9 x squared end cell row cell d t equals open parentheses negative 6 minus 18 x close parentheses d x end cell end table close vertical bar equals integral fraction numerator 1 over denominator square root of t end fraction d t equals integral t to the power of negative 1 half end exponent d t equals
equals fraction numerator 1 over denominator negative begin display style 1 half end style plus 1 end fraction t to the power of negative 1 half plus 1 end exponent plus C equals fraction numerator 1 over denominator begin display style 1 half end style end fraction t to the power of 1 half end exponent plus C equals 2 square root of t plus C equals 2 square root of 2 minus 6 x minus 9 x squared end root plus C

      I subscript 2 equals integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x

      Sprowadzamy dwumian w mianowniku do postaci a x squared plus b x plus c equals a open square brackets open parentheses x plus fraction numerator b over denominator 2 a end fraction close parentheses squared minus fraction numerator triangle over denominator 4 a squared end fraction close square brackets.

      triangle equals open parentheses negative 6 close parentheses squared minus 4 times open parentheses negative 9 close parentheses times 2 equals 36 plus 72 equals 108

      I subscript 2 equals integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x equals integral fraction numerator 6 over denominator square root of negative 9 open square brackets open parentheses x plus begin display style fraction numerator negative 6 over denominator 2 times open parentheses negative 9 close parentheses end fraction end style close parentheses squared minus begin display style fraction numerator 108 over denominator 4 times open parentheses negative 9 close parentheses squared end fraction end style close square brackets end root end fraction d x equals
equals integral fraction numerator 6 over denominator square root of negative 9 open square brackets open parentheses x plus begin display style 1 third end style close parentheses squared minus begin display style 1 third end style close square brackets end root end fraction d x equals integral fraction numerator 6 over denominator square root of negative 9 open parentheses x plus 1 third close parentheses squared plus 3 end root end fraction d x equals open vertical bar table row cell t equals x plus 1 third end cell row cell d t equals d x end cell end table close vertical bar equals
equals integral fraction numerator 6 over denominator square root of negative 9 t squared plus 3 end root end fraction d t equals integral fraction numerator 6 over denominator square root of 3 minus 9 t squared end root end fraction d t equals integral fraction numerator 6 over denominator square root of 9 open parentheses begin display style 3 over 9 end style minus t squared close parentheses end root end fraction d t equals integral fraction numerator 6 over denominator square root of 9 square root of 1 third minus t squared end root end fraction d t equals
equals integral fraction numerator 6 over denominator 3 square root of open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses squared minus t squared end root end fraction d t equals 2 integral fraction numerator 1 over denominator square root of open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses squared minus t squared end root end fraction d t equals 2 a r c sin fraction numerator t over denominator fraction numerator 1 over denominator square root of 3 end fraction end fraction plus C equals
equals 2 a r c sin fraction numerator square root of 3 t over denominator 2 end fraction plus C equals 2 a r c sin open square brackets square root of 3 open parentheses x plus 1 third close parentheses close square brackets plus C

      Podstawiamy I subscript 1 comma I subscript 2 tam, gdzie urwaliśmy, i mamy wynik:

      horizontal ellipsis equals negative 1 over 18 open parentheses stack stack integral fraction numerator negative 6 minus 18 x over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 1 below plus stack stack integral fraction numerator 6 over denominator square root of 2 minus 6 x minus 9 x squared end root end fraction d x with underbrace below with I subscript 2 below close parentheses equals
equals negative 1 over 18 open parentheses 2 square root of 2 minus 6 x minus 9 x squared end root plus 2 a r c sin open square brackets square root of 3 open parentheses x plus 1 third close parentheses close square brackets close parentheses plus C equals
equals negative 1 over 9 square root of 2 minus 6 x minus 9 x squared end root minus 1 over 9 a r c sin open square brackets square root of 3 open parentheses x plus 1 third close parentheses close square brackets plus C equals

       

  27. Kuba pisze:

    Witam, posiadam całkę(cos(x/5))/((sin^8(x/5)jak ją obliczyć metodą przez podstawianie t= sin(x/5)

  28. Marlena pisze:

    co oznacza ei

  29. Całka246 pisze:

    Dzien dobry, mam problem z rozwiązaniem poniższej całki… Nie mam kompletnie pomysłu jak się za to zabrać, próbowałam przez części jak i przed podstawienie, ale niestety do nieczego mądrego nie doszłam… integral subscript 1 superscript infinity fraction numerator x squared over denominator square root of x to the power of 7 plus 3 x plus 1 end root end fraction d x5c/ec/663032fd101522adc9baa9487e94.png” alt=”negative c space ln open parentheses c minus v close parentheses” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mi»c«/mi»«mo»§#160;«/mo»«mi»ln«/mi»«mfenced»«mrow»«mi»c«/mi»«mo»-«/mo»«mi»v«/mi»«/mrow»«/mfenced»«/math»” />Natomiast książka z książki od mechaniki, wynosi on negative c ln open parentheses 1 minus x over c close parenthesesCzy istnieje jakiś sposób na doprowadzenie z jednego wzoru do drugiego i ja tego nie widzę? Gdzie jest błąd?6b/aa/e5ff7f7200b2935730cfc4755204.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«/mfenced»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»ln«/mi»«mn»2«/mn»«mo»§#160;«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” /> 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  30. Mirek pisze:

    Czy ktoś ma pomysł jak rozwiązać poniższą całkę? Męczę się już dość długo z nią i wszystkie pomysły na jej rozwiązanie doprowadzają mnie do ślepego zaułku.integral open parentheses r squared over 2 a sin open parentheses fraction numerator x over denominator open vertical bar r close vertical bar end fraction close parentheses plus x over 2 square root of r squared minus x squared end root close parentheses cross times e to the power of negative x end exponent d x space space space g d z i e space r greater than 0 space i space r equals c o n s t
l u b space p r o ś c i e j
integral open parentheses a sin open parentheses x close parentheses plus x square root of 1 squared minus x squared end root close parentheses cross times e to the power of negative x end exponent d x5c/ec/663032fd101522adc9baa9487e94.png” alt=”negative c space ln open parentheses c minus v close parentheses” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mi»c«/mi»«mo»§#160;«/mo»«mi»ln«/mi»«mfenced»«mrow»«mi»c«/mi»«mo»-«/mo»«mi»v«/mi»«/mrow»«/mfenced»«/math»” />Natomiast książka z książki od mechaniki, wynosi on negative c ln open parentheses 1 minus x over c close parenthesesCzy istnieje jakiś sposób na doprowadzenie z jednego wzoru do drugiego i ja tego nie widzę? Gdzie jest błąd?6b/aa/e5ff7f7200b2935730cfc4755204.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«/mfenced»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»ln«/mi»«mn»2«/mn»«mo»§#160;«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” /> 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  31. Patryk pisze:

    Dzień dobry,mam problem z całką z funkcjifraction numerator 1 over denominator 1 minus begin display style x over c end style end fraction gdzie c to pewna stała. Wynik który wychodzi z kalkulatora to negative c space ln open parentheses c minus v close parenthesesNatomiast książka z książki od mechaniki, wynosi on negative c ln open parentheses 1 minus x over c close parenthesesCzy istnieje jakiś sposób na doprowadzenie z jednego wzoru do drugiego i ja tego nie widzę? Gdzie jest błąd?6b/aa/e5ff7f7200b2935730cfc4755204.png” alt=”integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mn»4«/mn»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«/mfenced»«mi»x«/mi»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«msqrt»«mn»1«/mn»«mo»-«/mo»«msup»«mfenced»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»ln«/mi»«mn»2«/mn»«mo»§#160;«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»ln«/mi»«mn»2«/mn»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” /> 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  32. BBB pisze:

    Witam, mam problem z całką (x^2)(sinx)^2 z zadania domowego nr 10 z lekcji 4. Wynik wychodzi mi podobny ale na końcu zamiast -(1/4)xcox2x+(1/8)sin2x mam -(1/2)xcos2x+(1/4)sin2x, tak jakbym coś zgubił po drodze, a liczyłem kilka razy i ciągle ten sam wynik. Ktoś coś?

  33. Sylwia pisze:

    Całka przez podstawienie : sin2x/square root of 3 minus sin to the power of 4 x end root67/90/94f2d0f705a7f949c05ba024c7f3.png” alt=”integral fraction numerator x cubed over denominator 1 plus x to the power of 8 end fraction d x equals integral fraction numerator x cubed over denominator 1 plus open parentheses x to the power of 4 close parentheses squared end fraction d x equals open vertical bar table row cell x to the power of 4 equals t end cell row cell 4 x cubed d x equals d t end cell row cell x cubed d x equals 1 fourth d t end cell end table close vertical bar equals integral 1 fourth fraction numerator 1 over denominator 1 plus t squared end fraction d t equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mi»x«/mi»«mn»8«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mfenced»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«/math»” />equals 1 fourth integral fraction numerator 1 over denominator 1 plus t squared end fraction d t equals 1 fourth a r c t g left parenthesis t right parenthesis plus C equals 1 fourth a r c t g left parenthesis x to the power of 4 right parenthesis plus C

    26) integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C

    3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  34. Sylwia pisze:

    Przez podstawienie: całka x^2/square root of 4 minus x hat 6 end rootdx67/90/94f2d0f705a7f949c05ba024c7f3.png” alt=”integral fraction numerator x cubed over denominator 1 plus x to the power of 8 end fraction d x equals integral fraction numerator x cubed over denominator 1 plus open parentheses x to the power of 4 close parentheses squared end fraction d x equals open vertical bar table row cell x to the power of 4 equals t end cell row cell 4 x cubed d x equals d t end cell row cell x cubed d x equals 1 fourth d t end cell end table close vertical bar equals integral 1 fourth fraction numerator 1 over denominator 1 plus t squared end fraction d t equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mi»x«/mi»«mn»8«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mfenced»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«/math»” />equals 1 fourth integral fraction numerator 1 over denominator 1 plus t squared end fraction d t equals 1 fourth a r c t g left parenthesis t right parenthesis plus C equals 1 fourth a r c t g left parenthesis x to the power of 4 right parenthesis plus C

    26) integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals 
    integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C

    3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  35. Sylwia pisze:

    Jak obliczyć całkę: 5x^2-6x+12/x^4-2x^3+4x^2

  36. ewcia5665 pisze:

    pomocy całka 4^x/2^x

    1. Tutaj wyjdzie ostatecznie prosta całka, trzeba tylko na samym początku dokonać kilku przekształceń:

       integral 4 to the power of x over 2 to the power of x d x equals integral open parentheses 2 squared close parentheses to the power of x over 2 to the power of x d x equals integral open parentheses 2 to the power of x close parentheses squared over 2 to the power of x d x equals integral 2 to the power of x space d x equals fraction numerator 2 to the power of x over denominator ln 2 end fraction plus C

      67/90/94f2d0f705a7f949c05ba024c7f3.png” alt=”integral fraction numerator x cubed over denominator 1 plus x to the power of 8 end fraction d x equals integral fraction numerator x cubed over denominator 1 plus open parentheses x to the power of 4 close parentheses squared end fraction d x equals open vertical bar table row cell x to the power of 4 equals t end cell row cell 4 x cubed d x equals d t end cell row cell x cubed d x equals 1 fourth d t end cell end table close vertical bar equals integral 1 fourth fraction numerator 1 over denominator 1 plus t squared end fraction d t equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mi»x«/mi»«mn»8«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mfenced»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«mo»§#8747;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»d«/mo»«mi»t«/mi»«mo»=«/mo»«/math»” />equals 1 fourth integral fraction numerator 1 over denominator 1 plus t squared end fraction d t equals 1 fourth a r c t g left parenthesis t right parenthesis plus C equals 1 fourth a r c t g left parenthesis x to the power of 4 right parenthesis plus C

      26) integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals 
      integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C

      3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

      negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  37. Piotr pisze:

    Proszę o rozpisanie tych całek są to przykłady z kursu pana Krystiana. Piszę ponieważ nie mogę ich rozgryźć i dojść do zgodności z wynikiem końcowym.Całka nieoznaczonaintegral open parentheses 4 minus 2 x close parentheses squared x d xcałka nieoznaczona przez podstawianieintegral fraction numerator ln to the power of 4 x over denominator x end fraction d x,  integral fraction numerator x cubed over denominator 1 plus x to the power of 8 end fraction d xintegral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fractionZa pomoc z góry dziękuję i pozdrawiam81/c6/e811718221afa031a01fa852f24d.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />.

    Rozwiązując dostaniemy

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table

    8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    1. ewcia5665 pisze:

      integral left parenthesis 4 minus 2 x right parenthesis squared equals space integral left parenthesis 16 minus 16 x plus 4 x squared right parenthesis space x space d x equals space integral left parenthesis 16 x minus 16 x squared plus 4 x cubed right parenthesis d x equals space integral 16 x d x
minus integral 16 x squared d x space plus integral 4 x cubed d x equals space 16 integral x space d x minus 16 integral x to the power of 2 space end exponent d x plus 4 integral x to the power of 3 to the power of blank end exponent d x equals 16 asterisk times space 1 divided by 2 space x squared minus space
16 asterisk times 1 divided by 3 space x to the power of 3 space end exponent plus 4 asterisk times space 1 divided by 4 space x to the power of 4 equals space 8 x squared minus 16 divided by 3 x cubed plus x to the power of 4e5/9d/72278c6d3c86c5bedbd1ef990b22.png” alt=”integral fraction numerator ln to the power of 4 x over denominator x end fraction d x” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfrac»«mrow»«msup»«mi»ln«/mi»«mn»4«/mn»«/msup»«mi»x«/mi»«/mrow»«mi»x«/mi»«/mfrac»«mi»d«/mi»«mi»x«/mi»«/math»” />,  integral fraction numerator x cubed over denominator 1 plus x to the power of 8 end fraction d xintegral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fractionZa pomoc z góry dziękuję i pozdrawiam81/c6/e811718221afa031a01fa852f24d.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” />.

      Rozwiązując dostaniemy

      table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table

      8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

      negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    2. Pozostałe całki:

      20)  integral fraction numerator ln to the power of 4 x over denominator x end fraction d x equals integral ln to the power of 4 x times 1 over x d x equals open vertical bar table row cell ln x equals t end cell row cell 1 over x d x equals d t end cell end table close vertical bar equals integral t to the power of 4 space d t equals 1 fifth t to the power of 5 plus C equals 1 fifth ln to the power of 5 x plus C

      25)   integral fraction numerator x cubed over denominator 1 plus x to the power of 8 end fraction d x equals integral fraction numerator x cubed over denominator 1 plus open parentheses x to the power of 4 close parentheses squared end fraction d x equals open vertical bar table row cell x to the power of 4 equals t end cell row cell 4 x cubed d x equals d t end cell row cell x cubed d x equals 1 fourth d t end cell end table close vertical bar equals integral 1 fourth fraction numerator 1 over denominator 1 plus t squared end fraction d t equalsequals 1 fourth integral fraction numerator 1 over denominator 1 plus t squared end fraction d t equals 1 fourth a r c t g left parenthesis t right parenthesis plus C equals 1 fourth a r c t g left parenthesis x to the power of 4 right parenthesis plus C

      26) integral fraction numerator 2 to the power of x over denominator square root of 1 minus 4 to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 squared close parentheses to the power of x end root end fraction d x equals integral fraction numerator 2 to the power of x over denominator square root of 1 minus open parentheses 2 to the power of x close parentheses squared end root end fraction d x equals open vertical bar table row cell 2 to the power of x equals t end cell row cell 2 to the power of x ln 2 space d x equals d t end cell row cell 2 to the power of x d x equals fraction numerator 1 over denominator ln 2 end fraction d t end cell end table close vertical bar equals 
      integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction fraction numerator 1 over denominator ln 2 end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction integral fraction numerator 1 over denominator square root of 1 minus t squared end root end fraction d t equals fraction numerator 1 over denominator ln 2 end fraction a r c sin t plus C equals fraction numerator 1 over denominator ln 2 end fraction a r c sin open parentheses 2 to the power of x close parentheses plus C

      3b/63/4af3d4beb1e1ccc2060e0c2a4ae0.png” alt=”table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
      table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»12«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»/«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mo»§#8230;«/mo»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«msup»«mrow/»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«mspace linebreak=¨newline¨/»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo» «/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»§#177;«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»” />8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

      negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

      Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

      T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

      T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

      9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

      T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

      T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

      integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  38. Maja Szczęsna pisze:

    Dzień dobry…. bardzo proszę o pomoc w rozwiązaniu całki …:cosx/(sinx^3+sinx^2)… 🙁

  39. Terazka pisze:

    Dzień dobry 🙂 Wstałam jaki sposób obliczyć całkę z (1+x)^x

  40. Maciej pisze:

    Witam mam pytanie jak obliczyc calke x^2/(3x^2-6)^(1/3)

  41. adek pisze:

    Witam,jak rozwinąć całkę od A (x^2)dA ?

  42. Popek pisze:

    integral fraction numerator d x over denominator cos cubed x end fraction Ktoś pomoże?20/ac/4db1e525480d4bff0b9a13e2c856.png” alt=”integral subscript square root of 2 end subscript superscript x fraction numerator 1 over denominator t square root of t squared minus 1 end root end fraction d t equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open square brackets a r c t g open parentheses square root of t squared minus 1 end root close parentheses close square brackets end cell end table subscript square root of 2 end subscript superscript x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of open parentheses square root of 2 close parentheses squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«msqrt»«mn»2«/mn»«/msqrt»«mi»x«/mi»«/msubsup»«mfrac»«mn»1«/mn»«mrow»«mi»t«/mi»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«mo»=«/mo»«msubsup»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mi»r«/mi»«mi»c«/mi»«mi»t«/mi»«mi»g«/mi»«mfenced»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«msqrt»«mn»2«/mn»«/msqrt»«mi»x«/mi»«/msubsup»«mo»=«/mo»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right 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columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mn»1«/mn»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” /> 

    gdzie w ostatniej równości skorzystano z faktu, że a r c t g left parenthesis 1 right parenthesis equals straight pi over 4 space b o space t g open parentheses straight pi over 4 close parentheses equals 1.

    Tym samy  równanie 

    integral subscript square root of 2 end subscript superscript x fraction numerator 1 over denominator t square root of t squared minus 1 end root end fraction d t equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table 

    sprowadzamy do postaci 

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table.

    Rozwiązując dostaniemy

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table

    8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

  43. Ola pisze:

    integral x square root of 2 minus x squared end root x d Błagam o pomoc .. xC 20/ac/4db1e525480d4bff0b9a13e2c856.png” alt=”integral subscript square root of 2 end subscript superscript x fraction numerator 1 over denominator t square root of t squared minus 1 end root end fraction d t equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open square brackets a r c t g open parentheses square root of t squared minus 1 end root close parentheses close square brackets end cell end table subscript square root of 2 end subscript superscript x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of open parentheses square root of 2 close parentheses squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table” align=”middle” data-mathml=”«math 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columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” /> 

    gdzie w ostatniej równości skorzystano z faktu, że a r c t g left parenthesis 1 right parenthesis equals straight pi over 4 space b o space t g open parentheses straight pi over 4 close parentheses equals 1.

    Tym samy  równanie 

    integral subscript square root of 2 end subscript superscript x fraction numerator 1 over denominator t square root of t squared minus 1 end root end fraction d t equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table 

    sprowadzamy do postaci 

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table.

    Rozwiązując dostaniemy

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table

    8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    1. klim pisze:

      t=2-x^2dt=2xdxdt/2=xdx

  44. Ola pisze:

    Witam mam problem z całką 1/x(x^2-2)^1/2 dx. Jak ją rozwiązać?

  45. Mateusz pisze:

    Proszę o pomoc część już zrobiłem ale potem się zatrzymałemintegral fraction numerator a r c t g x over denominator x squared end fraction equals open vertical bar table row cell f left parenthesis x right parenthesis space equals space a r c t g x end cell cell f apostrophe left parenthesis x right parenthesis space equals space fraction numerator 1 over denominator 1 plus x squared end fraction end cell row cell g apostrophe left parenthesis x right parenthesis space equals space 1 over x squared end cell cell g left parenthesis x right parenthesis space equals space minus 1 over x end cell end table close vertical bar equals negative fraction numerator a r c t g x over denominator x end fraction plus integral fraction numerator 1 over denominator x left parenthesis 1 plus x squared right parenthesis end fraction d xNie mogę rozgryźć tej całki ostatniej. Z góry dzięki  20/ac/4db1e525480d4bff0b9a13e2c856.png” alt=”integral subscript square root of 2 end subscript superscript x fraction numerator 1 over denominator t square root of t squared minus 1 end root end fraction d t equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open square brackets a r c t g open parentheses square root of t squared minus 1 end root close parentheses close square brackets end cell end table subscript square root of 2 end subscript superscript x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of open parentheses square root of 2 close parentheses squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
    table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table” align=”middle” data-mathml=”«math 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columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«mn»1«/mn»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd/»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»a«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»r«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»c«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mi»g«/mi»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfenced»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mo»-«/mo»«/mtd»«/mtr»«/mtable»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd/»«mtd/»«mtd»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»” /> 

    gdzie w ostatniej równości skorzystano z faktu, że a r c t g left parenthesis 1 right parenthesis equals straight pi over 4 space b o space t g open parentheses straight pi over 4 close parentheses equals 1.

    Tym samy  równanie 

    integral subscript square root of 2 end subscript superscript x fraction numerator 1 over denominator t square root of t squared minus 1 end root end fraction d t equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table 

    sprowadzamy do postaci 

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table.

    Rozwiązując dostaniemy

    table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 12 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank r end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared minus 1 end root close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank divided by end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses horizontal ellipsis close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of x squared minus 1 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell blank squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table
table attributes columnalign right center left columnspacing 0px end attributes row blank blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table

    8b/2b/fba86d83fbaea4a7d719d365b536.png” alt=”negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»cos«/mi»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi»ln«/mi»«mn»2«/mn»«mo»§#183;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»§#183;«/mo»«mi»sin«/mi»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«/math»” />

    negative 1 over 9 ln squared 2 times integral 2 to the power of x times sin 3 x d x

    Niech integral 2 to the power of x times sin 3 x d x equals T. Wtedy

    T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x minus 1 over 9 ln squared 2 times T. Stąd:

    T plus 1 over 9 ln squared 2 times T equals negative 1 third times 2 to the power of x times cos 3 x plus 1 over 9 ln 2 times 2 to the power of x times sin 3 x space divided by times 9

    9 T plus ln squared 2 times T equals negative 3 times 2 to the power of x times cos 3 x plus ln 2 times 2 to the power of x times sin 3 x space rightwards double arrow

    T times open parentheses 9 plus ln squared 2 close parentheses equals 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses space rightwards double arrow

    T equals fraction numerator 2 to the power of x times open parentheses negative 3 cos 3 x plus ln 2 times sin 3 x close parentheses over denominator 9 plus ln squared 2 end fraction space rightwards double arrow

    integral 2 to the power of x times sin 3 x d x equals fraction numerator negative 3 cos 3 x plus ln 2 times sin 3 x over denominator 9 plus ln squared 2 end fraction times 2 to the power of x plus C

    1. Mateusz pisze:

      Już rozwiązałem 🙂 

  46. Marlena pisze:

    Witam, proszę o pomoc w wyliczeniu jednej całki:integral open parentheses fraction numerator x squared minus 1 over denominator open parentheses x plus 1 close parentheses open parentheses x squared plus 1 close parentheses end fraction close parenthesescb/69/4bdec290c8827ab20e689c9ffe8e.png” alt=”open vertical bar table row cell sin x equals t end cell row cell cos x d x equals d t end cell end table close vertical bar equals” align=”middle” data-mathml=”«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«mi»sin«/mi»«mi»x«/mi»«mo»=«/mo»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»cos«/mi»«mi»x«/mi»«mo»d«/mo»«mi»x«/mi»«mo»=«/mo»«mo»d«/mo»«mi»t«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo»=«/mo»«/math»” />

    integral fraction numerator 1 minus t squared over denominator t cubed end fraction d t equals integral 1 over t cubed d t minus integral t squared over t cubed d t equals integral t to the power of negative 3 end exponent d t minus integral fraction numerator d t over denominator t end fraction equals

    fraction numerator t to the power of negative 3 plus 1 end exponent over denominator negative 3 plus 1 end fraction minus ln open vertical bar t close vertical bar plus C equals