1. 8.2 Integration By Parts Badlands, South Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

2. Integrate the following:

3. 8.2 Integration By Parts Start with the product rule: This is the Integration by Parts formula.

4. The Integration by Parts formula is a “product rule” for integration. u differentiates to zero (usually). dv is easy to integrate. Choose u in this order: LIPET Logs, Inverse trig, Polynomial, Exponential, Trig Or LIPTE

5. Integration by Parts !

6. Example 1: polynomial factor LIPET

7. Example 2: logarithmic factor LIPET

8. This is still a product, so we need to use integration by parts again. Example 3: LIPET

9. Example 4: LIPET This is the expression we started with!

10. Example 4 (con’t):

11. This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever.

12. Integration by Parts

13. A Shortcut: Tabular Integration Tabular integration works for integrals of the form: where: Differentiates to zero in several steps. Integrates repeatedly. Such as:

14. Compare this with the same problem done the other way:

15. Example 5: LIPET This is easier and quicker to do with tabular integration!

16. 

17. Find You Try:

18. Solution: Begin as usual by letting u = x 2 and dv = v' dx = sin 4x dx. Next, create a table consisting of three columns, as shown.

19. Homework: Day 1: pg. 531, 11-55 EOO, 59-69 odd. Day 2: MMM BC pgs. 106-107