More U-Substitution

1. More U-Substitution Chapter 5.5 February 8, 2007

2. In-Class Assignment • Integrate using two different methods: • 1st by multiplying out and integrating • 2nd by u-substitution Do you get the same result? (Don’t just assume or claim you do; multiply out your results to show it!) If you don’t get exactly the same answer, is it a problem? Why or why not?

3. Fundamental Theorem of Calculus (Part 2) • If f is continuous on [a, b], then : Where F is any antiderivative of f. ( )

4. Substitution Rule for Indefinite Integrals • If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Substitution Rule for Indefinite Integrals • If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

5. Using the Chain Rule, we know that: Evaluate:

6. Using the Chain Rule, we know that: Evaluate: Looks almost like cos(x2) 2x, which is the derivative of sin(x2).

7. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

8. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

9. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

10. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match: We put in a 2 so the pattern will match.

11. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match: We put in a 2 so the pattern will match. So we must also put in a 1/2 to keep the problem the same.

12. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

13. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

14. Check Answer:

15. Check Answer: Check:

16. Check Answer: Check: From the chain rule

17. Indefinite Integrals by Substitution

18. Indefinite Integrals by Substitution 1) Choose u.

19. Indefinite Integrals by Substitution Choose u. Calculate du.

20. Indefinite Integrals by Substitution Choose u. Calculate du. Substitute u.Arrange to have du in your integral also.(All xs and dxs must be replaced!)

21. Indefinite Integrals by Substitution Choose u. Calculate du. Substitute u.Arrange to have du in your integral also.(All xs and dxs must be replaced!) Solve the new integral.

22. Indefinite Integrals by Substitution Choose u. Calculate du. Substitute u.Arrange to have du in your integral also.(All xs and dxs must be replaced!) Solve the new integral. Substitute back in to get x again.

23. Example A linear substitution: Let u = 3x + 2. Then du = 3dx.

24. Choosing u • Try to choose u to be an inside function. (Think chain rule.) • Try to choose u so that du is in the problem, except for a constant multiple.

25. Choosing u For u = 3x + 2 was a good choice because 3x + 2 is inside the exponential. The derivative is 3, which is only a constant.

26. Practice Let u = du = 

27. Practice Let u = x2 + 1du = 

28. Practice Let u = x2 + 1du = 2xdx

29. Practice Let u = x2 + 1du = 2xdx

30. Practice Let u = x2 + 1du = 2xdx Make this a 2xdx and we’re set!

31. Practice Let u = x2 + 1du = 2xdx

32. Practice Let u = x2 + 1du = 2xdx

33. Practice Let u = x2 + 1du = 2xdx

34. Practice Let u = x2 + 1du = 2xdx

35. Practice Let u = du = 

36. Practice Let u = sin(x)du = 

37. Practice Let u = sin(x)du = cos(x) dx

38. Practice Let u = sin(x)du = cos(x) dx

39. Practice Let u = sin(x)du = cos(x) dx

40. Practice Let u = sin(x)du = cos(x) dx

41. Practice Let u = du =  An alternate possibility:

42. Practice Let u = cos(x)du = –sin(x) dx An alternate possibility:

43. Practice Let u = cos(x)du = –sin(x) dx An alternate possibility:

44. Practice Let u = cos(x)du = –sin(x) dx An alternate possibility:

45. Practice Let u = cos(x)du = –sin(x) dx An alternate possibility:

46. Practice Let u = cos(x)du = –sin(x) dx An alternate possibility:

47. Practice Note:

48. Practice Note:

49. Practice Note: What’s the difference?

50. Practice Note: What’s the difference?