Warm-up:

1. Warm-up: Evaluate the integrals. 1) 2)

2. Warm-up: Evaluate the integrals. 1) 2)

3. Warm-up: Evaluate the integrals. 1) 2)

4. Integration by Parts Section 8.2 Objective: To integrate problems without a u-substitution

5. Integration by Parts • When integrating the product of two functions, we often use a u-substitution to make the problem easier to integrate. Sometimes this is not possible. We need another way to solve such problems.

6. Integration by Parts • As a first step, we will take the derivative of

7. Integration by Parts • As a first step, we will take the derivative of

8. Integration by Parts • As a first step, we will take the derivative of

9. Integration by Parts • As a first step, we will take the derivative of

10. Integration by Parts • As a first step, we will take the derivative of

11. Integration by Parts • Now lets make some substitutions to make this easier to apply.

12. Integration by Parts • This is the way we will look at these problems. • The two functions in the original problem we are integrating are u and dv. The first thing we will do is to choose one function for u and the other function will be dv.

13. Example 1 • Use integration by parts to evaluate

14. Example 1 • Use integration by parts to evaluate

15. Example 1 • Use integration by parts to evaluate

16. Example 1 • Use integration by parts to evaluate

17. Example 1 • Use integration by parts to evaluate

18. Guidelines • The first step in integration by parts is to choose u and dv to obtain a new integral that is easier to evaluate than the original. In general, there are no hard and fast rules for doing this; it is mainly a matter of experience that comes from lots of practice.

19. Guidelines • There is a useful strategy that may help when choosing u and dv. When the integrand is a product of two functions from different categories in the following list , you should make u the function whose category occurs earlier in the list. • Logarithmic, Inverse Trig, Algebraic, Trig, Exponential • The acronym LIATE may help you remember the order.

20. Guidelines • If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.

21. Guidelines • If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.

22. Guidelines • Since the new integral is harder than the original, we made the wrong choice.

23. Example 2 • Use integration by parts to evaluate

24. Example 2 • Use integration by parts to evaluate

25. Example 2 • Use integration by parts to evaluate

26. Example 2 • Use integration by parts to evaluate

27. Example 2 • Use integration by parts to evaluate

28. Example 3 (S): • Use integration by parts to evaluate

29. Example 3 • Use integration by parts to evaluate

30. Example 3 • Use integration by parts to evaluate

31. Example 3 • Use integration by parts to evaluate

32. Example 3 • Use integration by parts to evaluate

33. Example 4 (Repeated): • Use integration by parts to evaluate

34. Example 4 (Repeated): • Use integration by parts to evaluate

35. Example 4 (Repeated): • Use integration by parts to evaluate

36. Example 4 (Repeated): • Use integration by parts to evaluate

37. Example 4 (Repeated): • Use integration by parts to evaluate

38. Example 4 (Repeated): • Use integration by parts to evaluate

39. Example 4 (Repeated): • Use integration by parts to evaluate

40. Example 4 (Repeated): • Use integration by parts to evaluate

41. Example 5: • Evaluate the following definite integral

42. Example 5: • Evaluate the following definite integral

43. Example 5: • Evaluate the following definite integral

44. Example 5: • Evaluate the following definite integral

45. Example 5: • Evaluate the following definite integral

46. Example 5: • Evaluate the following definite integral

47. Example 5: • Evaluate the following definite integral

48. Example 5: • Evaluate the following definite integral

49. Example 5: • Evaluate the following definite integral

50. Example 5: • Evaluate the following definite integral