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   Change of Variable
Let F(x) be an indefinite integral or antiderivative of f(x). Then
change of variable in indefinite integral
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 .

Solution.
Let . Then . Hence, the integral is
     
   Example 2
Find the integral .

Solution.
We make the substitution . Then or .
The integral is easy to calculate with the new variable:
     
   Example 3
Calculate the integral .

Solution.
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 .

Solution.
We can write the integral as
     
Changing the variable
     
we get the answer
     
   Example 5
Find the integral .

Solution.
We make the following substitution:
     
Hence,
     

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