1. Objective: To integrate functions using a u-substitution

2. U-Substitution
The method of substitution can be motivated by examining the chain rule from the viewpoint of antidifferentiation. For this purpose, suppose that F is an antiderivative of f and that g is a differentiable function. The chain rule implies that the derivative of F(g(x)) can be expressed as
which we can write in integral form as

3. U-Substitution
For our purposes it will be useful to let u = g(x) and to write in the differential form
With this notation, our formula becomes

4. Example 1
Evaluate

5. Example 1
Evaluate
We will let Most times the function being raised to an exponent will be u.

6. Example 1
Evaluate
We will let Most times the function being raised to an exponent will be u.
If then We will solve for dx, so

7. Example 1
Evaluate
We will let Most times the function being raised to an exponent will be u.
If then We will solve for dx, so
We will now write the integral as

8. Example 1
Evaluate
We will now integrate the new function and then substitute back in for u.

9. Guildelines
If something is being raised to an exponent (including a radical), that will be u.
If one function is 1 degree higher than the other function, that will be u.
If e is being raised to an exponent, that exponent will be u.
If you have one trig function, the inside function will be u.

10. Example 2
Evaluate

11. Example 2
Evaluate

12. Example 2
Evaluate

13. Example 2
Evaluate

14. Example 2
Evaluate

15. Example 2
Evaluate

16. Example 2
Evaluate

17. Example 3
Evaluate

18. Example 3
Evaluate

19. Example 3
Evaluate

20. Example 4
Evaluate

21. Example 4
Evaluate

22. Example 4
Evaluate

23. Example 4
Evaluate

24. Example 5
Evaluate

25. Example 5
Evaluate

26. Example 5
Evaluate

27. Example 5
Evaluate

28. Example 6
Evaluate

29. Example 6
Evaluate

30. Example 6
Evaluate

31. Example 6
Evaluate

32. Example 6
Evaluate

33. Example 7
Evaluate

34. Example 7
Evaluate

35. Example 7
Evaluate

36. Example 7
Evaluate

37. Example 8
Evaluate

38. Example 8
Evaluate

39. Example 8
Evaluate

40. Example 9
Evaluate

41. Example 9
Evaluate

42. Example 9
Evaluate

43. Example 9
Evaluate

44. Example 10
Evaluate

45. Example 10
Evaluate

46. Example 10
Evaluate

47. Example 10
Evaluate

48. Example 11
Evaluate

49. Example 11
Evaluate

50. Example 11
Evaluate

51. Example 11
Evaluate

52. Example 11
Evaluate