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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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Volume of a Solid of Revolution – Problem with a Catch

24 April 2024 / Krystian Karczyński

Learn how to easily calculate the volume of a solid of revolution using a simple formula from elementary school. Discover the method through an example of calculating the volume of a sphere formed by rotating a curve around an axis, without complex integrals.

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Definite Integrals Calculated with Respect to the Variable y

22 April 2024 / Krystian Karczyński

Definite integrals can be calculated with respect to variable x or y, and sometimes should be, if it's more convenient. This often plays a big role in the applications of integrals, such as calculating the areas of regions, arc lengths, volumes, and surface areas of solids of revolution. Often we don't even have a choice, because the conditions of the problem specify that the curve rotates around the OY axis, not the OX. How do you do that?

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Definite Integrals by Substitution – Changing the Limits of Integration

22 April 2024 / Krystian Karczyński

Discover how to effectively change integration limits by substitution in definite integral tasks. This article explains in detail the step-by-step process of changing limits, using real examples and formulas to simplify understanding of this crucial mathematical technique.

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Gaussian vs Cramer’s vs Kronecker-Capelli Methods – Matrices in Solving Systems of Linear Equations

22 April 2024 / Krystian Karczyński

Discover how Gauss, Cramer, and Kronecker-Capelli methods help in efficiently solving systems of linear equations. Learn why Gauss's method is recommended for larger systems of equations, and compare the advantages and limitations of each method.

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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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Function Derivatives – How to Start?

19 April 2024 / Krystian Karczyński

To start learning how to calculate function derivatives (the real ones, not from the definition), you should of course start by preparing a sheet with formulas. What next? I'll give you a tip.

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Fourth Degree Polynomial Equations in Complex Numbers

18 April 2024 / Krystian Karczyński

When solving complex polynomial equations, we generally use the same methods as for solving real polynomial equations in high school. How to handle fourth-degree polynomials?

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Trigonometric form of a complex number – different angles, same solution?

18 April 2024 / Krystian Karczyński

Trigonometric functions (used in the trigonometric form of a complex number) are 2π-periodic, which means that for any arguments differing by e.g., 2π, 4π, 6π, they assume the same value.

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