Indefinite Rational Integrals – Third-Degree Polynomial in the Denominator

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Krystian Karczyński

In indefinite rational integrals, as we know, we often need to factor the denominator of the integrand and further decompose it into simple fractions.

However, factorization itself can often be tricky.

Typical situations

For simple situations like:

We can handle it:

If the denominator is a properly “arranged” third-degree polynomial, we can still manage:

Denominator with an unfriendly third-degree polynomial

The problem starts when the denominator has a more “unfriendly” third-degree polynomial to factorize, for example:

Deep breath.

Control the panic.

Such things (factoring this type of polynomial) were done in high school and at the basic level.

Just need a clever trick:

Remember? Of course you do…

So we can factorize the polynomial in the denominator:

And then cheerfully decompose it into three simple fractions, as the rules tell us.

Even more difficult factorization

And what about such a situation? Remember from high school (this is more of an extended scope)?

Here it was a bit worse. The middle term needed to be split into two:

And then proceed as usual:

So the integral should be written as:

And continue calculating calmly.

Methods of factorizing the denominator in rational integrals

Generally, remember in rational integrals that if factorizing the polynomial is needed, all high school tricks are allowed, and they might include, for example:

  • Factorizing some fourth-degree polynomials by substitution
  • Blindly searching for a root and dividing the polynomial to reduce its degree

Of course, we are now only talking about the more difficult indefinite rational integrals, which in many (most?) universities are not even introduced!

konometria jest dosyć młodą dziedziną wypływającą z ekonomii i matematyki. W praktyce, dzięki modelom ekonometrycznym, możesz „zmierzyć gospodarkę”.Polega to konkretnie na zmierzeniu, jak zachowuje się jedna zmienna w zależności od innych. I na podstawie analizy tego, co było, możesz określać, co będzie się działo w przyszłości.

Wykorzystasz do tego przeróżne obliczenia, testy, schematy. Jedne będą bardzo proste, inne trudniejsze. Jednak najczęściej będzie się liczyło nie to, jak dojdziesz do wyniku, ale jak go zinterpretujesz, odczytasz i jakie wnioski wyciągniesz.

Poniższe Wykłady dotykają najważniejszych pojęć teoretycznych. Jestem przekonana, że pomogę Ci odkrywaniu tego, czym jest ekonometria. I przy okazji uda Ci się zaliczyć ten przedmiot na studiach.

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