1. Warm Up

2. Sect. 4.5 Integration
By Substitution

3. Chain Rule for Derivatives:
Chain Rule backwards for Integration:
This means that in order to integrate a composite function, the derivative of the “inside” must be present, and you integrate the “outside.”

4. Back to Warm Up 1 Let u = 3x – 2x2, then du/dx = 3 – 4x

5. Back to Warm Up 2 du
Let u = 1 – 2x2,
then du/dx = – 4x
du = – 4x dx

6. u – substitution in Integration
Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

7. Recognizing the Pattern
Given each integral, determine u = g(x) and du = g’(x)dx.
g’(x)
g(x)
g’(x)
(g(x))1/2
(g(x))2
g(x) g’
g(x)
g’(x)
g(x)
eg(x)
cos(g(x))

8. Making a Change of Variables
The steps used for integration by substitution are summarized in the following guidelines.
1. Choose a substitution u = g(x). Usually it is best to choose the inner part of a composite function.
2. Compute du = g’(x) dx
3. Rewrite the integral in terms of the variable u.
4. Evaluate the resulting integral in terms of u.
5. Replace u by g(x) to obtain an antiderivative in terms of x.
6. Check your answer by differentiating.

9. Remember you may differentiate to check your work!
Ex 1:
u = x2 + 1
Remember you may differentiate to check your work!
Substitute, simplify and integrate:

10. Ex 2:
u = 5x
Substitute, simplify and integrate:

11. Ex 3: Find
Let
Substitute, simplify and integrate:

12. Ex 4: Find
Substitute into the integral, simplify and integrate.

13. You Try…
Evaluate each of the following.

14. Example 1 Solution u1 du
u = x2 + x
u2/2
Example 3 Solution

15. Multiplying/Dividing by a Constant
Ex 5:
Let u = x2 + 1
du = 2x dx
problem!
Rewrite original with factor of 2

16. Ex 6:

17. Warning! The constant multiple rule only applies to constants!!!!!
You CANNOT multiply and divide by a variable and then move the variable outside the integral.

18. You Try…
Evaluate each of the following. 6.

19. Example 4 Solution
Let u =
4x3 then du =
12x2 dx
Example 5 Solution
Let u =
5 – 2x2 then du =
- 4x dx

20. Example 6 Solution

21. More Complicated Examples
u = sin 3x
du = 3cos 3x

22. Example 8
Evaluate

23. You Try…
Evaluate each of the following.

24. Example 1 Solution

25. Example 9 Find

26. Example 10

27. Example 11

28. You Try…
Evaluate each of the following.

29. Example 6 Solution

30. Example 12

31. Example 13

32. Example 14

33. Example 15

34. Warm Up
Evaluate

35. Shortcuts: Integrals of Expressions Involving ax + b

36. Back to Warm Up
Evaluate

37. Evaluate each of the following. 1. 2. 3.
You Try…
Evaluate each of the following. 1. 2. 3.

38. For some integrals, the shortcut does not apply
For some integrals, the shortcut does not apply. Instead, FOIL and integrate using the Power Rule. 

39. Definite Integrals

40. Ex 1:
u = x2 + 1
du = 2x dx
Or, we could use new limits if we leave u in.
Calculate new upper and lower limits by substituting the old ones in for x in u.
x = 0  u = 1
x = 1  u = 2
Notice

41. You Try…
Sketch the region represented by each definite integral then evaluate. Use a calculator to verify your answer. 1. 2.

42. One option is to change the limits
1 Solution
One option is to change the limits
u = 3t Then when t = 1, u = when t = 2, u = 5
Resulting integral

43. 2 Solution
Don’t forget to use the new limits.

44. Ex 2:

45. Using the original limits:
Ex 2:
Using the original limits:
Leave the limits out until you substitute back.

46. Ex 2:
Using the new limits:
new limit
new limit

47. You Try…
Sketch the region represented by each definite integral then evaluate. Use a calculator to verify your answer. 3.

48. 3 Solution

49. You Try… 4.

50. Definite Integrals & Transformations
Given , find
a. b.
c d.
e. f.

51. You Try…
Given , find
a. b.
c d.
e. f.

52. Even/Odd Functions Given the graph, evaluate the
definite integral of
Given the graph and ,
evaluate

53. Even – Odd Property of Integrals

54. Symmetric about the y axis
FUNCTIONS
Symmetric about the origin

55. A function is even if f( -x) = f(x) for every number x in the domain.
So if you plug a –x into the function and you get the original function back again it is even.
Is this function even?
YES
Is this function even? NO

56. A function is odd if f( -x) = - f(x) for every number x in the domain.
So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd.
Is this function odd? NO
Is this function odd?
YES

57. If a function is not even or odd we just say neither (meaning neither even nor odd)
Determine if the following functions are even, odd or neither.
Not the original and all terms didn’t change signs, so NEITHER.

58. Ex 3: a) Evaluate
Find the area of the region
bounded by the x-axis and the curve

59. Example 4: Evaluate Let f(x) = x4 – 29x2 + 100
Since f(x) = f(-x), f is an
even function.

60. Since f(x) = f(-x), f is an odd function.
Ex 5: Evaluate
Let f(x) = sin3x cos x + sin x cos x
f(–x) = sin3(–x) cos (–x) + sin (–x) cos (–x)
= –sin3x cos x – sin x cos x = –f(x)
Since f(x) = f(-x), f is an odd function.

61. You Try…
Find the area of the region bounded by the x-axis and the curve of on the interval [-2, 2].

62. You Try… -a a

63. u-sub with a Twist Ex 1: problem! Substitute in terms of u
Antiderivative in terms of u
Antiderivative in terms of x

64. Ex 2: problem! Substitute in terms of u Antiderivative in terms of u
Antiderivative in terms of x

65. Alternate u
Ex 2:

66. Ex 3:
Evaluate

67. You Try…

68. Example 2 Solution
u = 2x du = 2 dx

69. determine new upper and lower limits of integration
Back to Ex 2: as a definite integral
determine new upper and lower limits of integration
When x = 1, u = 1
lower limit
upper limit
x = 5, u = 9

70. Ex 2: (cont.)
Before substitution
After substitution =
Area of region is 16/3
Area of region is 16/3

71. Example 4

72. You Try…

73. Example 3 Solution
u = x du = dx

74. Using Long Division Before Integrating
Back to Example 3

75. Example 5

76. You Try…

77. Evaluate each of the following. 1. 2.
Warm Up
Evaluate each of the following. 1. 2.

78. Integrals of Trig Functions
A useful example of integration by substitution is to find
Ex 1:
Let
Using a law of logs,

79. A little trickier one…
First, multiply numerator and denominator by sec x + tan x.
u = sec x + tan x, du = (sec x tan x + sec2x)
∫ (1/u) du = ln |u| + C
Ex 2:

80. Integrals of Trig Functions

81. Shortcuts: Integrals involving (ax + b)

82. Examples: 3.
4. Evaluate

83. Sect. 5.9 Inverse Trig Functions: Integration

84. Integrals of Inverse Trig Functions
If each integral sign above had a – in front, we would get an antiderivative of the corresponding co-function.

85. Ex 1
Ex 2

86. Ex 3
Ex 4

87. You Try…
Evaluate each of the following.

88. Ex 1 Solution

89. Ex 5

90. You Try… Evaluate each of the following.
Hint: Split up into two separate fractions first.

91. You Try…
5. Find the area enclosed by x = -¼, x = ¼, y = 0, and

92. Warning
Many integrals look like the inverse trig forms, but require a different technique.
Which of the following are of the inverse trig forms? If they are not, how are they integrated?