Substitution Rule

1. Substitution Rule

2. Substitution Rule Basic Problems

3. Example (1)

6. Example (2)

9. Example (3)

12. Example (4)

14. Example (5)

16. Example (6)

18. Substitution Rule Definite Integral Case

19. Example (1)

22. Example (2)

25. Example (3)

28. Substitution Rule More Challenging Problems

29. Example (1)

30. Method 1

36. Note that the first method can be used to find the integral of any function of the form:f(x) = x(2n-1) (axn+b)kfor any positive integer n and any real number k (where k is not -1) as the following examples show:

37. Example (2)

38. In all of the first three examples, we let:u = 2x+ 4and so:du = 2dx → dx = du/2andx = (u - 4)/2

42. In the fourth example, we let:u = 2x2+ 4and so:du = 4xdx → dx = du/4xandx2 = (u - 4)/2

44. In the fifth example, we let:u = 2x3+ 4and so:du = 6x2dx → dx = du/6x2andx3 = (u - 4)/2

46. Examples (3)

47. The double angle formulas can simplify these problems, by replacing cos2x by (1+cos2x)/2 and sin2x by (1- cos2x)/2