1. Integration by Substitution (Section 4-5)

2. 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.

3. The variable of integration must match the variable in the expression.
Find the indefinite integral.
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!

4. Find the indefinite integral.
Example 2

5. Find the indefinite integral.
Example 3

6. Find the indefinite integral.
Example 4:
Solve for dx.

7. Find the indefinite integral.
Example 5:

8. Note that this only worked because of the 2x in the original.
Find the indefinite integral.
One of the clues that we look for is if we can find a function and its derivative in the integrand.
Example 6:
The derivative of is
Note that this only worked because of the 2x in the original.
Many integrals can not be done by substitution.

9. We solve for because we can find it in the integrand.
Find the indefinite integral.
Example 7:
We solve for because we can find it in the integrand.

10. Find the indefinite integral.
Example 8:

11. Example 9: Find the indefinite integral by the method shown in Example 5.

12. HW #1 pg 304 (1-33odd)