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Indeterminate integrals

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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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How do I know what to substitute in indefinite integrals?

17 April 2024 / Krystian Karczyński

Learn how to effectively use the substitution method in solving indefinite integrals. This article contains detailed tips and examples that will help you understand when and which substitutions to choose to simplify difficult integrals. Ideal for mathematics students starting their journey in integration, looking for practical advice and solutions.

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Euler Substitutions of the First Kind

05 January 2012 / Krystian Karczyński

Euler Substitutions – Who Needs Them? Euler substitutions in indefinite integrals are the next topic introduced after rational integrals, trigonometric integrals, and integrals with roots (or according to some classifications: “irrational integrals”). This means that most students won’t have the pleasure of encountering them. I also didn’t include them in my Course on Indefinite Integrals. […]

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Calculator For Indeterminate Integrals (Check If You’re Counting Right).

27 April 2012 / Krystian Karczyński

Introducing the Wolfram calculator for indeterminate integrals, reworked a bit by me.

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