4.5: Integration by Substitution Objectives: Assignment:

4.5: Integration by Substitution
Objectives:
Assignment:
1. To use a change of
variables to evaluate
indefinite and definite
integrals
• P. 304-305: 7-33 odd, 3747 odd, 61-67 odd, 70,
71-81 eoo, 83, 87
2. To use symmetry as an
aid in integration
• P. 306: 101-107 odd
Warm Up 1
Find the derivative of ℎ 𝑥 = 𝑥 2 + 𝑥.
ℎ 𝑥 = 𝑥2 + 𝑥
ℎ′ 𝑥 =
ℎ′ 𝑥 =
1/2
1 2
𝑥 +𝑥
2
2𝑥 + 1
−1/2
2𝑥 + 1
𝑔′ 𝑥
2 𝑥2 + 𝑥
𝑔 𝑥
𝑓 𝑥 = 𝑥
The Chain Rule
If 𝑦 = 𝑓 𝑢 is a differentiable function of 𝑢 and
𝑢 = 𝑔 𝑥 is a differentiable function of 𝑥, then
𝑦 = 𝑓 𝑔 𝑥 is a differentiable function of 𝑥
and
𝑑𝑦 𝑑𝑦 𝑑𝑢
=
∙
𝑑𝑥 𝑑𝑢 𝑑𝑥
𝑑
𝑓 𝑔 𝑥
𝑑𝑥
= 𝑓′ 𝑔 𝑥
∙ 𝑔′ 𝑥
The derivative of the exterior function times
the derivative of the interior function
Warm Up 2
2𝑥+1
′
Find the antiderivative of ℎ 𝑥 =
ℎ′ 𝑥 =
2𝑥 + 1
2
𝑥 2 +𝑥
𝑔′ 𝑥
2 𝑥2 + 𝑥
𝑓 𝑥 = 𝑥
𝑔 𝑥
ℎ′ 𝑥 = 𝑓′ 𝑔 𝑥 𝑔′ 𝑥
ℎ 𝑥 =𝑓 𝑔 𝑥
+𝐶
=
𝑥2 + 𝑥 + 𝐶
.
Objective 1
You will be able to
use a change of
variables to
evaluate indefinite
and definite
integrals
Composite Functions
When integrating a composite function 𝑓,
realize that function 𝑓 was the product of the
Chain Rule.
𝐹′ 𝑔 𝑥
𝑔′ 𝑥 𝑑𝑥 = 𝐹 𝑔 𝑥
+𝐶
Antidifferentiation of Composite Functions
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
𝑓 𝑔 𝑥
𝑔′ 𝑥 𝑑𝑥 =
𝐹 𝑔 𝑥
+𝐶
Substitution Rule
If 𝑢 = 𝑔(𝑥), then
𝑑𝑢
𝑑𝑥
= 𝑔′ 𝑥 .
𝑑𝑢 = 𝑔′ 𝑥 𝑑𝑥
𝑓 𝑔 𝑥
𝑔′ 𝑥 𝑑𝑥 =
𝑓 𝑢 𝑑𝑢 = 𝐹 𝑢 + 𝐶
This substitution is accomplished with a change of
variable, where 𝑢 is a function of 𝑥.
Exercise 1
Find
1 + 𝑥 2 2𝑥 𝑑𝑥
2𝑥 1 + 𝑥 2 𝑑𝑥 =
Let 𝑢 = 1 + 𝑥 2
𝑑𝑢
= 2𝑥
𝑑𝑥
=
Always remember
that your answer
should be in terms of
the original variable.
𝑑𝑢 = 2𝑥 𝑑𝑥
𝑢 𝑑𝑢 =
𝑢1/2 𝑑𝑢
2 3/2
2
= 𝑢 + 𝐶 = 1 + 𝑥2
3
3
3/2
+𝐶
Check your answer
by taking its
derivative.
Protip
When changing variables, try letting 𝑢 =
Some complicated part of the integrand
The inner function of a composite function
Some function whose derivative is part of the integrand
Use Constant Multiple Rule
Except for a constant factor
If your first
guess doesn’t
work, try
something
different.
Exercise 2
Find
2𝑥 𝑥 2 + 1
2
𝑑𝑥.
Exercise 3
Find
𝑥 𝑥2 + 1
2
𝑑𝑥.
Exercise 4
Find
5cos 5𝑥 𝑑𝑥.
Exercise 5
Find
2𝑥 − 1 𝑑𝑥.
Exercise 6
Find
𝑥 2𝑥 − 1 𝑑𝑥.
Protip:
When changing
variables,
sometimes you
have to use the
equation for 𝑢 to
solve for 𝑥 to do
another
substitution.
Exercise 7
Find
sin2 3𝑥 cos 3𝑥 𝑑𝑥.
General Power Rule for Integration
If 𝑔 is a differentiable function of 𝑥, then
𝑛+1
𝑔
𝑥
𝑔 𝑥 𝑛 𝑔′ 𝑥 𝑑𝑥 =
+ 𝐶,
𝑛+1
If 𝑢 = 𝑔 𝑥 , then
𝑛+1
𝑢
𝑢𝑛 𝑑𝑢 =
+ 𝐶,
𝑛+1
𝑛 ≠ −1
𝑛 ≠ −1
Exercise 8
Find the following antiderivatives.
1.
3 3𝑥 − 4
4 𝑑𝑥
2.
2𝑥 + 1 𝑥 2 + 𝑥
4 𝑑𝑥
Exercise 8
Find the following antiderivatives.
3.
3𝑥 2
𝑥3
− 2 𝑑𝑥
4.
−4𝑥
𝑑𝑥
1−2𝑥 2 2
Exercise 9
Find
cos 2 𝑥 sin 𝑥 𝑑𝑥.
Exercise 10
Evaluate
1
𝑥
0
𝑥2 + 1
3
𝑑𝑥.
Method 1:
𝑢 = 𝑥2 + 1
𝑑𝑢 = 2𝑥 𝑑𝑥
𝑑𝑢
= 𝑥 𝑑𝑥
2
𝑏
𝑎
𝑑𝑢
1
3
𝑢
=
2
2
𝑏
𝑎
1 4
3
𝑢 𝑑𝑢 = 𝑢
8
𝑏
𝑎
1 2
= 𝑥 +1
8
=
15
8
1
4
0
Exercise 10
Evaluate
1
𝑥
0
𝑥2 + 1
3
𝑑𝑥.
Method 2:
𝑢 = 𝑥2 + 1
𝑑𝑢 = 2𝑥 𝑑𝑥
𝑑𝑢
= 𝑥 𝑑𝑥
2
2
1
𝑑𝑢
1
𝑢3
=
2
2
New Limits of Integration:
𝑢 = 12 + 1 = 2
𝑢 = 02 + 1 = 1
2
1
1
𝑢3 𝑑𝑢 = 𝑢4
8
15
=
8
2
1
Definite Integrals
If the function
𝑢 = 𝑔 𝑥 has a
continuous
derivative on the
𝑎, 𝑏 and 𝑓 is
continuous on
the range of 𝑔,
then
𝑏
𝑔 𝑏
𝑓 𝑔 𝑥 𝑔′ 𝑥 𝑑𝑥 =
𝑎
𝑓 𝑢 𝑑𝑢
𝑔 𝑎
Exercise 11
Evaluate 𝐴 =
4
0
2𝑥 + 1 𝑑𝑥.
Objective 2
You will be able to use
symmetry as an aid in
integration
Exercise 12
Evaluate each definite integral.
1.
2 6
𝑥
−2
+ 1 𝑑𝑥
2.
2 6
𝑥
0
+ 1 𝑑𝑥
Exercise 13
Evaluate
𝜋/3
sin
3𝑥
𝑑𝑥.
−𝜋/3
Symmetry
Recall that the graphs of even and odd functions
have a useful kind of symmetry:
Even = 𝑦-axis Symmetry
𝑓 −𝑥 = 𝑓 𝑥
Odd = Origin Symmetry
𝑓 −𝑥 = −𝑓 𝑥
Integration of Even/Odd Functions
Let 𝑓 be integrable on −𝑎, 𝑎 .
If 𝑓 is an even function, then
𝑎
𝑎
𝑓
𝑥
𝑑𝑥
=
2
𝑓 𝑥 𝑑𝑥.
−𝑎
0
If 𝑓 is an odd function, then
𝑎
𝑓 𝑥 𝑑𝑥 = 0.
−𝑎
Exercise 14
Evaluate
𝜋/2
3
sin
𝑥
cos
𝑥
−𝜋/2
+ sin 𝑥 cos 𝑥 𝑑𝑥.
4.5: Integration by Substitution
Objectives:
Assignment:
1. To use a change of
variables to evaluate
indefinite and definite
integrals
• P. 304-305: 7-33 odd, 3747 odd, 61-67 odd, 70,
71-81 eoo, 83, 87
2. To use symmetry as an
aid in integration
• P. 306: 101-107 odd