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How do I know what to substitute in indefinite integrals?

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.

The Biggest Problem with Matrix Problems…

Learn effective methods to avoid computational errors while calculating the rank and determinant of matrices. Discover how to properly conduct matrix operations to minimize the risk of mistakes and optimize your solutions.

The Derivative of the Square Root of x – How to Derive the Formula

In this post, I show how to derive the formula for the derivative of the square root of x from the definition of a derivative. Briefly, precisely, and step by step.

Euler Substitutions of the First Kind

Euler's substitutions in indefinite integrals are the next thing introduced after rational integrals, trigonometric integrals, and root integrals. This means that most students won't have the pleasure of encountering them, and I haven't included them in my Course on Indefinite Integrals.

However, there remains quite a large group of students in mathematical fields who have to deal with Euler's substitutions, and I invite these (and the curious) to join. I will discuss all three types of Euler's substitutions (in this post I will cover the first type) and provide one example for each.

Homogeneous systems of linear equations (number of solutions using matrix rank)

Homogeneous systems of linear equations are systems in which all intercepts are equal to 0. I will show you how to solve such systems of equations using a row of matrices.

Horizontal and oblique asymptotes of functions

The relationship between horizontal and oblique asymptotes is as follows: horizontal asymptotes are special oblique asymptotes. Therefore, every horizontal asymptote is an oblique asymptote, but not every oblique is horizontal. This can and should be used to shorten the counting of function asymptotes.

Two types of discontinuity (function limits)

As we all know (for example from my Limits Course), a function is continuous at the point x0 when the left-side limit of this function at this point is equal to the right-side limit of the function at this point and is equal to the value of the function at this point. If any of the equality is not satisfied, the function f(x) is not continuous at the point x0, and the point is called a point of discontinuity. In this naming, you can go a step further and DISTINGUISH the points of discontinuity. See how to do it.

Proving that sinx does not reach a limit as x approaches infinity

We have a limit of the function as x approaches infinity from the function f(x)=sinx. We intuitively feel that the above limit does not exist. x's are getting bigger and bigger, and the sine values ​​are constantly fluctuating between -1 and 1. But how can we formally demonstrate and prove this?

Useful limit of a function with cosine

We usually treat the limits of a sine function as: x tends to zero then{sinx} /x=1. Question: what about cosine of x? Does it have any "typical" border?