4.5 Integration by Substitution The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.

Example 1: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem!

Example 2: One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is . Note that this only worked because of the 2x in the original. Many integrals can not be done by substitution.

Example 3: Solve for dx.

Example 4:

Example 5 We solve for because we can find it in the integrand.

Example 6:

Evaluating Definite Integrals

The technique is a little different for definite integrals. Example 1: The technique is a little different for definite integrals. new limit We can find new limits, and then we don’t have to substitute back. new limit

Example 2: Don’t forget to use the new limits.