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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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Elliptical Coordinates (Double Integrals)

03 July 2024 / Krystian Karczyński

There are times in life when the region of integration in a double integral is an ellipse.... What do we do then?

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Helpful tricks in complex numbers

02 July 2024 / Krystian Karczyński

Complex numbers as a whole are not a complicated and difficult topic. However, things can get "heated" in less typical and less schematic situations. The key then, as always, is to understand the topic and keep a "cool head," meaning being clear-headed and confident.

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The (Almost) Trigonometric Form of a Complex Number

02 July 2024 / Krystian Karczyński

When solving problems with complex numbers, it's important to keep in mind that a complex number in trigonometric form has its typical form. And only that form. No more, no less. Therefore, you need to pay attention to when a complex number is in trigonometric form and when it is not.

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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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Why can’t a function graph have two oblique asymptotes as x approaches positive infinity?

28 June 2024 / Krystian Karczyński

To answer the question posed in the title, we don't need to go back to the definition of an oblique asymptote, we just need to know what a function is.

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The Symbol 0 in the Limits of Functions and Sequences

27 June 2024 / Krystian Karczyński

When working with limits of functions and sequences, we frequently come across various mathematical oddities, such as: 0/0 or 1/0. The question is - what do they mean? I often come across completely incorrect opinions like: "at higher levels in mathematics, you can divide by 0".

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The Rank of a Matrix with a Parameter

26 June 2024 / Krystian Karczyński

Check out the article where I show how to calculate the rank of a matrix with one example.

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The Limit of a Sequence with Two Mixed Sums of Arithmetic Sequences

26 June 2024 / Krystian Karczyński

Sequence limits sometimes use formulas for the sum of arithmetic or geometric sequences. It's worse if these sequences are given in a "mixed" way in one line, like here.

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