1. Integration by Substitution Lesson 5.5

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

3. 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 …

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

5. Example Try this …  what is the g(u)?  what is the du/dx? We have a problem … Where is the 4 which we need?

6. Example We can use one of the properties of integrals We will insert a factor of 4 inside and a factor of ¼ outside to balance the result Where did the 1/3 come from? Why is this now a 3?

7. Can You Tell? Which one needs substitution for integration? Go ahead and do the integration.

8. Try Another …

9. Assignment A Lesson 5.5 Page 340 Problems: 1 – 33 EOO 49 – 77 EOO

10. Change of Variables We completely rewrite the integral in terms of u and du Example: So u = 2x + 3 and du = 2x dx But we have an x in the integrand  So we solve for x in terms of u

11. Change of Variables We end up with It remains to distribute the and proceed with the integration Do not forget to "un-substitute"

12. What About Definite Integrals Consider a variation of integral from previous slide One option is to change the limits  u = 3t - 1 Then when t = 1, u = 2 when t = 2, u = 5  Resulting integral

13. What About Definite Integrals Also possible to "un-substitute" and use the original limits

14. Integration of Even & Odd Functions Recall that for an even function  The function is symmetric about the y-axis Thus An odd function has  The function is symmetric about the orgin Thus

15. Assignment B Lesson 5.5 Page 341 Problems: 87 - 109 EOO 117 – 132 EOO