Indeterminate integrals

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Hyperbolic Functions to the Rescue – My New Lecture on the Blog

16 August 2024 / Krystian Karczyński

I wrote a new Lecture entirely dedicated to hyperbolic functions and their inverses. In it, I explain what hyperbolic functions are, when - for example - they are useful, and why they sometimes appear in Wolfram results, generally causing panic among students. However, the hyperbolic sine isn't that scary - I invite you to the Lecture.

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Forbidden Formulas for Indefinite Integrals – Derivation of Formulas

08 August 2024 / Krystian Karczyński

Professors at universities have their own requirements. Many of them, for the good of their students, of course, will not hesitate to specify the rules for solving problems in great detail. Some do not recognize ready-made formulas for some indefinite integrals. See where they came from.

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Euler Substitutions of the Third Kind – Summary

02 August 2024 / Krystian Karczyński

In previous posts, I demonstrated how to use Euler substitutions in integrals with the square root of the polynomial ax^2+bx+c. Euler substitutions of the first kind were used when a>0, and second kind Euler substitutions when c>0. In this post, we will deal with the third and final kind of Euler substitutions that can be used when the quadratic polynomial in the integral has TWO DISTINCT roots x1, x2, meaning its discriminant is positive. See what to do in this case.

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Euler Substitution of the Second Kind

30 July 2024 / Krystian Karczyński

In the previous post: Euler Substitution of the First Kind, we dealt with integrals involving the root of the trinomial {ax^2+bx+c}, where a>0. But what if "a" in the trinomial is negative? Then the second kind of Euler substitution, for c>0, might help us (but not necessarily...).

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Canonical Form of a Quadratic Function in Rational Integrals

03 July 2024 / Krystian Karczyński

I get a lot of questions about the formula for the canonical form of a quadratic function introduced in my Course on Indefinite Integrals. Why is there a^2? Let me explain.

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Indefinite Rational Integrals – Third-Degree Polynomial in the Denominator

28 June 2024 / Krystian Karczyński

In indefinite rational integrals, as we know, it is often necessary to factorize the denominator of the integrand and further decompose it into simple fractions. However, factorization itself can often be tricky.

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Delta equals zero in rational indefinite integrals

29 April 2024 / Krystian Karczyński

In rational indefinite integrals, there’s often a need to factorize the quadratic trinomial ax^2 + bx + c. We do this using the formula ax^2 + bx + c = a(x - x1)(x - x2), which works when Delta is greater than 0. But what happens when Delta is precisely 0?

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What squared equals 17? Namely, indefinite integrals and the formula with arctan

29 April 2024 / Krystian Karczyński

We've gotten used to applying the formula with arctan tgx in "clean" integral situations. What if there are expressions that seem to have no "neat" root? Discover methods for solving indefinite integrals with the arctan function.

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Different outcomes, correct solution, in indefinite integral tasks – possible?

22 April 2024 / Krystian Karczyński

Wondering why you get different results from those listed in the textbook when solving indefinite integrals? Learn how different integration methods and adding a constant can affect the outcomes and how to correctly interpret these differences.

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