Lecture 8 – Integration Basics

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

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. 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 Likewise,by parts is the “almost” reverse of the product rule. u-substitution is the reverse of the chain rule.

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

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

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. 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) Trig Substitution

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 U 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