CHAPTER 4 SECTION 4.5 INTEGRATION BY SUBSTITUTION

1. CHAPTER 4SECTION 4.5INTEGRATION BY SUBSTITUTION

2. Theorem 4.12 Antidifferentiation of a Composite Function

3. Substitution with Indefinite Integration This is the “backwards” version of the chain rule Recall … Then …

4. Substitution with Indefinite Integration • In general we look at the f(x) and “split” it • into a g(u) and a du/dx • So that …

5. Substitution with Indefinite Integration Note the parts of the integral from our example

6. Substitution with Indefinite Integration Let u = So, du = (2x -4)dx

7. Guidelines for Making a Change of Variables

8. Theorem 4.13 The General Power Rule for Integration

9. 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!

10. 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 2:

11. Example 3: Solve for dx.

12. Example 4:

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

14. Example 6:

15. Can You Tell? • Which one needs substitution for integration?

16. Integration by Substitution

17. Integration by Substitution

22. Solve the differential equation

23. Solve the differential equation

24. Theorem 4.14 Change of Variables for Definite Integrals

26. or you could convert the bound to u’s.

27. The technique is a little different for definite integrals. new limit new limit Example 7: We can find new limits, and then we don’t have to substitute back.

28. Example 9: Don’t forget to use the new limits.

29. Theorem 4.15 Integration of Even and Odd Functions

30. Even/Odd Functions If f(x) is an even function, then If f(x) is an odd function, then

31. Even/Odd Functions If f(x) is an even function, then If f(x) is an odd function, then