4. 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.

5. 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!

6. 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.

7. Example 3:
Solve for dx.

8. Example 4:

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

10. Example 6:

11. Evaluating Definite
Integrals

12. 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

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