1. 4.5 (part 1) Integration by Substitution
M.L.King Jr. Birthplace, Atlanta, GA
Greg Kelly
Hanford High School
Richland, Washington
Photo by Vickie Kelly, 2002

2. Objectives Use pattern recognition to evaluate an indefinite integral.
Use a change of variables to evaluate an indefinite integral.
Use the General Power Rule for Integration to evaluate an indefinite integral.

3. Do you remember the Chain Rule?
If , then
Backwards Chain Rule!

4. Sometimes this “backwards chain rule” can be accomplished using u-substitution.
A good rule of thumb is to let
u = something that’s raised to a power
or the “inside function”.

5. Example:
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:
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:
The derivative of is

8. Example:
Solve for dx.

9. Example:
Solve for dx.

10. Example:

11. Example:

12. Example:

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

14. Example:

15. If you had to solve one of these, which would you choose?

16. Homework
4.5 (page 306)
# 1 – 45 odd
(43 & 45 ignore directions)