1. Lecture 8 – Integration Basics
Functions – know their shapes and properties
A few (very few) examples:

2. Trigonometric Rules
Know basics about sine, cosine, tangent, secant, plus 2 right triangles
Beyond these angles:
and use reference angles for all quadrants.

3. Substitution Rule
First approach for any integral should be a u-substitution. Ex. 1 Which (if any) of the following can use a basic u-substitution?

4. Ex. 2 Which (if any) of the following can use a basic u-substitution?

5. Know derivatives for trig functions. Ex
Know derivatives for trig functions. Ex. 3 But what about antiderivatives?

6. Ex.4 What antiderivative for secant function?

7. Ex.5

8. Need to try a different 1.

9. Lecture 9 – Integration By Parts
U-substitution is the reverse of the chain rule.
Likewise, by parts is the “almost” reverse of the product rule.

10. When figuring out integrals, now looking for one of the following:
1: know the
2: look for
3: look for
4: look for
When trying to decide what to use for the u,

11. Example 1

12. Example 2

13. Example 3

14. Example 4 What is needed to solve each?

15. Lecture 10 – More Integration By Parts
Example 5

16. Example 6 What is needed to solve each?

17. Example 7

19. Example 8

21. Lecture 11 – Trig Integrals
Use u-sub, trig identities, and/or by parts.

22. Trig identities:

23. Example 1

24. Example 2

25. Example 3 With dealing with sine or cosine functions,
you are looking for (cos x dx) or (sin x dx),
respectively.

26. Example 4

27. Lecture 12 – More Trig Integration
With dealing with tangent or secant, you are looking for
(sec2 x dx) or (sec x tan x dx), respectively.
Example 5

28. Example 6

29. Example 7

31. Trig Substitution
When faced with one of the above in an integral , create a
right triangle and substitute trig expressions in  for algebraic
expressions of x. (unless a simple u-substitution is available)

33. Lecture 13 –Trig Substitution
Example 1

35. Example 2

37. Example 3

39. Lecture 14 – Partial Fractions
Combine the following:

40. Process can be reversed so that any rational function can be
expressed as the sum of partial fractions.
Any polynomial can be rewritten as a product of linear and
irreducible quadratic factors. So q(x) can be decomposed.
Linear fractions have only a constant in the numerator,
regardless of the number of repetitions.
Quadratic fractions have linear and constant terms only.

41. Why useful? U S B T R I G S U B

42. Example 1

43. Example 2

45. Example 3

47. Example 4

49. Lecture 15 – Improper Integrals
Infinite Integrals: infinity at one or both limits.
Example 1

50. Example 2

51. Example 3

52. Example 4

53. Example 5
Find the volume of the solid generated by revolving the region bounded the curve and the x-axis on the interval around the x-axis.
f(x) x 1 2 3 2

54. x 2

55. Lecture 16 – More Improper Integrals
Discontinuous Integrands: infinite discontinuity in intervals [a, b].
Example 6

56. Example 7

57. Example 8

58. Example 9