The Method of Integration by Parts

1. The Method of Integration by Parts

2. Main Idea If u & v are differentiable functions of x, then By integrating with respect to x, we get:

3. When to use this method? When the integrand is a product of the form udv, such that we do not know how to find the integral ∫udv, but can find v = ∫dv and the integral ∫vdu.

4. Examples I When, we have an integrand, similar to one of the following: ( where b and c are any real numbers) • xn cos(cx) or xn sin(cx) ; where n is a natural number 2. xn ecx or xn acx ; where n is a natural number and b is a base for an exponential function ( b is positive and not equal to 1) 3. xlnx or xc lnxb

5. Example 1

7. Example 2

9. Example 3

11. Example 5

13. Example 5

15. Example 6

17. Example 7

19. Example 8

23. Example 9

25. Examples II:Integrals valued byRepeated Use of the Method When, we have an integrand, similar to one of the following: ( where b and c are any real numbers) • sin(bx) cos(cx) 2. ecx sin(bx) or ecx cos(bx)

26. Example 1

29. Example 2

32. Another method to evaluate this integral and similar ones is to use the proper trigonometric identitiesRecall that:

33. Using the identity (2), we get:

34. Home Quiz 1.Prove the identity(2) of the previously given trigonometric identities 2.Show that the two values arrived at for the integral of of this example are equivalent

35. Examples IIIUsing the method to find the integrals of trigonometric and inverse trigonometric functions that can not be found directly A. ∫arcsinx dx , ∫arccosx dx , ∫arctanx dx , ∫arccotx dx, ∫arcsecx dx and ∫arccscx dx B. ∫secnx dx and ∫cscnx dx , where n is an odd natural number greater than 1

36. Examples III - AExample 1

38. Example 2

40. Example 3

42. We find the last integral using the method of trigonometric substitution

44. Substituting that back, we get:

45. Examples III - BExample 1

48. Examples III - BExample 2