1. Basic Integration Rules Lesson 8.1

2. Fitting Integrals to Basic Rules Consider these similar integrals Which one uses … The log rule The arctangent rule The rewrite with long division principle

3. Try It Out Decide which principle to apply …

4. The Log Rule in Disguise Consider The quotient suggests possible Log Rule, but the _________ is not present We can manipulate this to make the Log Rule apply Add and subtract e x in the numerator

5. The Power Rule in Disguise Here's another integral that doesn't seem to fit the basic options What are the options for u ? Best choice is

6. The Power Rule in Disguise Then becomes and _____________applies Note review of basic integration rules pg 520 Note procedures for fitting integrands to basic rules, pg 521

7. Disguises with Trig Identities What rules might this fit? Note that tan 2 u is ____________________ However sec 2 u is on the list This suggests one of the _____________________identities and we have

8. Assignment Lesson 8.1 Page 522 Exercises 1 – 49 EOO

9. Integration by Parts Lesson 8.2

10. Review Product Rule Recall definition of derivative of the product of two functions Now we will manipulate this to get

11. Manipulating the Product Rule Now take the integral of both sides Which term above can be simplified? This gives us

12. Integration by Parts It is customary to write this using substitution u = f(x)du = ____________ v = g(x) _________ = g'(x) dx

13. Strategy Given an integralwe split the integrand into two parts First part labeled u The other labeled dv Guidelines for making the split The dv always includes the _______ The ______ must be integratable v du is ___________________________than u dv Note: a certain amount of trial and error will happen in making this split

14. Making the Split A table to keep things organized is helpful Decide what will be the _____ and the _____ This determines the du and the v Now rewrite udu dvv

15. Strategy Hint Trick is to select the correct function for u A rule of thumb is the LIATE hierarchy rule The u should be first available from L___________________ Inverse trigonometric A___________ Trigonometric E________________

16. Try This Given Choose a u and dv Determine the v and the du Substitute the values, finish integration udu dvv

17. Double Trouble Sometimes the second integral must also be done by parts udu dvv ux2x2 du2x dx dvsin xv-cos x

18. Going in Circles When we end up with the the same as we started with Try Should end up with Add the integral to both sides_____________

19. Application Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π What is the volume generated by rotating the region around the y-axis? What is the radius? What is the disk thickness? What are the limits? What is the radius? What is the disk thickness? What are the limits?

20. Assignment Lesson 8.2A Page 531 Exercises 1 – 35 odd Lesson 8.2B Page 532 Exercises 47 – 57, 99 – 105 odd

21. Trigonometric Integrals Lesson 8.3

22. Recall Basic Identities Pythagorean Identities Half-Angle Formulas These will be used to integrate powers of sin and cos

23. Integral of sin n x, n Odd Split into product of an __________________ Make the even power a power of sin 2 x Use the Pythagorean identity Let u = cos x, du = -sin x dx

24. Integral of sin n x, n Odd Integrate and un-substitute Similar strategy with cos n x, n odd

25. Integral of sin n x, n Even Use half-angle formulas Try Change to power of ________ Expand the binomial, then integrate

26. Combinations of sin, cos General form If either n or m is odd, use techniques as before Split the _____ power into an ________power and power of one Use Pythagorean identity Specify u and du, substitute Usually reduces to a ____________ Integrate, un-substitute

27. Combinations of sin, cos Consider Use Pythagorean identity Separate and use sin n x strategy for n odd

28. Combinations of tan m, sec n When n is even Factor out ______________ Rewrite remainder of integrand in terms of Pythagorean identity sec 2 x = _______________ Then u = tan x, du = sec 2 x dx Try

29. Combinations of tan m, sec n When m is odd Factor out tan x sec x (for the du) Use identity sec 2 x – 1 = tan 2 x for _________ powers of tan x Let u = ___________________, du = sec x tan x Try the same integral with this strategy Note similar strategies for integrals involving combinations of cot m x and csc n x

30. Integrals of Even Powers of sec, csc Use the identity sec 2 x – 1 = tan 2 x Try

31. Wallis's Formulas If n is odd and (n ≥ ___) then If n is even and (n ≥ ___) then These formulas are also valid if cos n x is replaced by _______

32. Wallis's Formulas Try it out …

33. Assignment Lesson 8.3 Page 540 Exercises 1 – 41 EOO

34. Trigonometric Substitution Lesson 8.4

35. New Patterns for the Integrand Now we will look for a different set of patterns And we will use them in the context of a right triangle Draw and label the other two triangles which show the relationships of a and x 35 a x

36. Example Given Consider the labeled triangle Let x = 3 tan θ(Why?) And dx = 3 sec 2 θ dθ Then we have 36 3 x θ

37. Finishing Up Our results are in terms of θ We must un-substitute back into x Use the ____________________ 37 3 x θ

38. Knowing Which Substitution 38 u u

39. Try It!! For each problem, identify which substitution and which triangle should be used 39

40. Keep Going! Now finish the integration 40

41. Application Find the arc length of the portion of the parabola y = 10x – x 2 that is above the x-axis Recall the arc length formula 41

42. Special Integration Formulas Useful formulas from Theorem 8.2 Look for these patterns and plug in the a 2 and u 2 found in your particular integral

43. Assignment Lesson 8.4 Page 550 Exercises 1 – 45 EOO Also 67, 69, 73, and 77 43

44. Partial Fractions Lesson 8.5

45. Partial Fraction Decomposition Consider adding two algebraic fractions Partial fraction decomposition ___________ the process

46. Partial Fraction Decomposition Motivation for this process The separate terms are __________________

47. The Process Given Where polynomial P(x) has ______________ P(r) ≠ 0 Then f(x) can be decomposed with this cascading form

48. Strategy Given N(x)/D(x) 1.If degree of N(x) _____________ degree of D(x) divide the denominator into the numerator to obtain Degree of N 1 (x) will be _________ that of D(x) Now proceed with following steps for N 1 (x)/D(x)

49. Strategy 2.Factor the denominator into factors of the form where is irreducible 3.For each factor the partial fraction must include the following sum of m fractions

50. Strategy 4.Quadratic factors: For each factor of the form, the partial fraction decomposition must include the following sum of n fractions.

51. A Variation Suppose rational function has distinct linear factors Then we know

52. A Variation Now multiply through by the denominator to clear them from the equation Let x = 1 and x = -1 (Why these values?) Solve for A and B

53. What If Single irreducible quadratic factor But P(x) degree < 2m Then cascading form is

54. Gotta Try It Given Then

55. Gotta Try It Now equate corresponding coefficients on each side Solve for A, B, C, and D ?

56. Even More Exciting When but P(x) and D(x) are polynomials with ___________________________ D(x) ≠ 0 Example

57. Combine the Methods Consider where P(x), D(x) have no common factors D(x) ≠ 0 Express as ____________functions of

58. Try It This Time Given Now manipulate the expression to determine A, B, and C

59. Partial Fractions for Integration Use these principles for the following integrals

60. Why Are We Doing This? Remember, the whole idea is to make the rational function easier to integrate

61. Assignment Lesson 8.5 Page 559 Exercises 1 – 45 EOO

62. Integration by Tables Lesson 7.1

63. Tables of Integrals Text has covered only limited variety of integrals Applications in real life encounter many other types _______________________to memorize all types Tables of integrals have been established Text includes list in Appendix B, pg A-18

64. General Table Classifications Elementary forms Forms involving Trigonometric forms Inverse trigonometric forms Exponential, logarithmic forms Hyperbolic forms

65. Finding the Right Form For each integral Determine the classification Use the given pattern to complete the integral

66. Reduction Formulas Some integral patterns in the tables have the form This reduces a given integral to the sum of a ______________ and a ______________integral Given Use formula 19 first of all

67. Reduction Formulas This gives you Now use formula 17 and finish the integration

68. Assignment Lesson 8.6 Page 565 Exercises 1 – 49 EOO

69. Indeterminate Forms and L’Hopital’s Rule Lesson 8.7

70. Problem There are times when we need to evaluate functions which are rational At a specific point it may evaluate to an indeterminate form

71. Example of the Problem Consider the following limit: We end up with the indeterminate form Note why this is indeterminate

72. L’Hopital’s Rule When gives an indeterminate form (and the limit exists) It is possible to find a limit by Note: this only works when the original limit gives an ________________ form

73. Example Consider As it stands this could be Must change to format So we manipulate algebraically and proceed

74. Example Consider Why is this not a candidate for l’Hospital’s rule?

75. Example Try When we apply l’Hospital’s rule we get We must apply the rule a _____________

76. Hints Manipulate the expression until you get one of the forms Express the function as a _________ to get

77. Assignment Lesson 8.7 Page 574 Exercises 1 – 57 EOO

78. Improper Integrals Lesson 7.7

79. Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral is Known as an improper integral

80. To Infinity and Beyond To solve we write as a limit (if the limit exists)

81. Improper Integrals Evaluating Take the integral Apply the limit

82. To Limit Or Not to Limit The limit may not exist Consider Rewrite as a limit and evaluate

83. To Converge Or Not For A limit exists (the proper integral converges) for _______________ The integral _________________ for p ≤ 1

84. Improper Integral to - Try this one Rewrite as a limit, integrate

85. When f(x) Unbounded at x = c When vertical asymptote exists at x = c Given As before, set a limit and evaluate In this case the limit is __________

86. Using L'Hopital's Rule Consider Start with integration by parts dv _______ and u = ______________ Now apply the definition of an improper integral

87. Using L'Hopital's Rule We have Now use _______________________for the first term

88. Assignment Lesson 8.8 Page 585 Exercises 1 – 61 EOO