Change of Variable

Let F(x) be an indefinite integral or antiderivative of f(x). Then where x = g (u) is a substitution. Accordingly, the inverse function u = g ⁻¹(x) describes the dependence of the new variable on the old variable.

It's important to remember that the differential dx also needs to be substituted. It must be replaced with the differential of the new variable du. For definite integrals, it is also necessary to change the limits of integration. See about this on the page "The Definite Integral and Fundamental Theorem of Calculus".

   Example 1

Calculate the integral . Let . Then . Hence, the integral is

   Example 2

Find the integral . We make the substitution . Then or .
The integral is easy to calculate with the new variable:

   Example 3

Calculate the integral . Rewrite the integral in the following way: Noting 2e = a (This is not a change of variable, since x still remains the independent variable), we get the table integral:

   Example 4

Calculate the integral . We can write the integral as Changing the variable we get the answer

   Example 5

Find the integral . We make the following substitution: Hence,