1. Review Of Formulas And Techniques Integration Table

2. EX 1.1 A Simple Substitution EX 1.2 Generalizing a Basic Integration Rule

3. EX 1.3 An Integrand That Must Be Expanded EX 1.4 An Integral Where We Must Complete the Square

4. EX 1.5 An Integral Requiring Some Imagination

5. Integration By Parts Let

6. ？ EX 2.1 Integration by Parts Let So Remark How about letting and

7. EX 2.3 An Integrand with a Single Term Let

8. EX 2.4 Repeated Integration by Parts

9. EX 2.5 Repeated Integration by Parts with a Twist

10. Reduction Formula EX 2.6 Using a Reduction Formula …

11. Integration by Parts with A Definite Integral EX 2.7 Integration by Parts for a Definite Integral

12. Integrals Involving Powers of Trigonometric Functions Case 1: m or n Is an Odd Positive Integer Case 2: m and n Are Both Even Positive Integers Type A

13. EX 3.2 An Integrand with an Odd Power of Sine Let

14. EX 3.3 An Integrand with an Odd Power of Cosine Let

15. EX 3.4 An Integrand with an Even Power of Sine EX 3.5 An Integrand with an Even Power of Cosine

16. Type B Integrals Involving Powers of Trigonometric Functions Case 1: m Is an Odd Positive Integer Case 2: n Is an Even Positive Integer Case 3: m Is an Even Positive Integer and n Is an Odd Positive Integer

17. EX 3.6 An Integrand with an Odd Power of Tangent

18. EX 3.7 An Integrand with an Even Power of Secant

19. EX 3.8 An Unusual Integral Evaluate

20. Trigonometric Substitution How to integrate the following forms ? for some a > 0 Case A Case B Case C

21. EX 3.9 An Integral Involving

22. EX 3.10 An Integral Involving

23. EX 3.11 An Integral Involving

24. Using Partial Fractions Distinct linear factors Repeated linear factors Irreducible quadratic factors

25. EX 4.1 Partial Fractions: Distinct Linear Factors    

26. EX 4.2 Partial Fractions: Three Distinct Linear Factors 

27. EX 4.3 Partial Fractions Where Long Division Is Required 

28. EX 4.4 Partial Fractions with a Repeated Linear Factor 

29. EX 4.5 Partial Fractions with a Quadratic Factor 

30. EX 4.6 Partial Fractions with a Quadratic Factor A=2 B=3 and C=4 

31. EX 4.6 (Cont’d) 

33. Reasons The Fundamental Theorem assumes a continuous integrand, our use of the theorem is invalid and our answer is incorrect. 1 2

35. EX 6.1 An Integrand That Blows Up at the Right Endpoint EX 6.2 A Divergent Improper Integral

36. EX 6.3 A Convergent Improper Integral

37. EX 6.4 A Divergent Improper Integral

39. EX 6.5 An Integrand That Blows Up in the Middle of an Interval

42. EX 6.6 An Integral with an Infinite Limit of Integration EX 6.7 A Divergent Improper Integral

43. EX 6.8 A Convergent Improper Integral

44. EX 6.9 An Integral with an Infinite Lower Limit of Integration EX 6.10 A Convergent Improper Integral

45. Remark It’s certainly tempting to write this, especially since this will often give a correct answer, with about half of the work. Unfortunately, this will often give incorrect answers, too, as the limit on the right-hand side frequently exists for divergent integrals.

46. EX 6.11 An Integral with Two Infinite Limits of Integration

47. EX 6.12 An Integral with Two Infinite Limits of Integration

48. EX 6.13 An Integral That Is Improper for Two Reasons f(x) is not continuous on [ 0,∞).

50. EX 6.14 Using the Comparison Test for an Improper Integral

51. EX 6.15 Using the Comparison Test for an Improper Integral Remark

52. EX 6.16 Using the Comparison Test: A Divergent Integral