Section 4.5 Integration by Substitution
IB Calculus
Objective:
Mrs. Francis
In this lesson you learn how to evaluate different types of
definite and indefinite integrals using a variety of methods.
Date:
I. Pattern Recognition (Pages 297299)
The role of substitution in integration is comparable to the role of
in differentiation.
𝒅
[𝑭(𝒈(𝒙))]
𝒅𝒙
= 𝑭′ (𝒈(𝒙))𝒈′(𝒙)
From the definition of an antiderivative, it follows that:
∫ 𝑭′ (𝒈(𝒙))𝒈′ (𝒙)𝒅𝒙 = 𝑭(𝒈(𝒙)) + 𝑪
Antidifferentiation of a Composite Function
∫ 𝑭′ (𝒈(𝒙))𝒈′ (𝒙)𝒅𝒙 = 𝑭(𝒈(𝒙)) + 𝑪
Letting 𝒖 = 𝒈(𝒙) gives 𝒅𝒖 = 𝒈′ (𝒙)𝒅𝒙 and
From the definition of an antiderivative, it follows that:
∫ 𝒇(𝒖)𝒅𝒖 = 𝑭(𝒖) + 𝑪
𝟐
Example 1:
∫(𝒙𝟐 + 𝟏) (𝟐𝒙)𝒅𝒙
Example 2:
∫ 𝟓 𝒄𝒐𝒔 𝟓𝒙 𝒅𝒙
What you should
learn How to use
pattern recognition
to find an indefinite
integral
Many integrands contain the essential part (the variable part) of
𝑔′(𝑥) but are missing a constant multiple. In such cases, you can
.
∫ 𝒌 𝒇(𝒙)𝒅𝒙 = 𝒌 ∫ 𝒇(𝒙)𝒅𝒙
Example 3: ∫ 𝒙(𝒙𝟐 + 𝟏)𝟐 𝒅𝒙
II. Change of Variables (Pages 300301)
What you should learn
You should learn how
to integrate using
substitution
With a formal change of variables, you completely
.

 f (g(x))g (x) dx 
.
Example 4:
a. Find ∫ √2𝑥 − 1 𝑑𝑥
b. Find ∫ 𝑠𝑖𝑛2 3𝑥 cos 3𝑥 𝑑𝑥
Guidelines For Making Change of Variables
1. Choose a substitution 𝑢 = 𝑔(𝑥). Usually it is best to choose the inner
part of a composite function, such as a quantity raised to a power.
2. Compute 𝑑𝑢 = 𝑔′ (𝑥)𝑑𝑥
3. Rewrite the integral in terms of the variable𝑢.
4. Find 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.
III. The General Power Rule for Integration
One of the most common u substitutions involves quantities in the integrand
that are raised to a power. Because of the importance of the type of substitution,
it is given a special name – the General Power Rule for Integration.
Example 5 – Substitution and the General Power Rule
a.
∫ 3(3𝑥 − 1)4 𝑑𝑥
b.
∫(2𝑥 + 1)(𝑥 2 + 𝑥)𝑑𝑥
c.
∫ 3𝑥 2 √𝑥 3 − 2 𝑑𝑥
d.
∫ (1−2𝑥 2)2 𝑑𝑥
e.
∫ 𝑐𝑜𝑠 2 𝑥 sin 𝑥 𝑑𝑥
−4𝑥
IV. Change of Variables for Definite Integrals
(Pages 303304)
When using u-substitution with a definite integral, it is often
convenient to
rather than to convert the antiderivative
back to the variable x and evaluate the original limits.
Change of Variables for Definite Integrals
If the function u g(x) has a continuous derivative on the
closed interval [a, b] and f is continuous on the range of g, then
Example 6 – Change of Variables
1
Evaluate ∫0 𝑥(𝑥 2 + 1)3 𝑑𝑥
What you should learn
How to use a change of
variables to evaluate a
definite integral
V. Integration of Even and Odd Functions (Page 305)
Occasionally, you can simplify the evaluation of a definite
integral over an interval that is symmetric about the y-axis or
about the origin by
.
Let f be integrable on the closed interval [a, a].
𝑎
𝑎
If f is an
function, then ∫−𝑎 𝑓(𝑥)𝑑𝑥 = 2 ∫0 𝑓(𝑥)𝑑𝑥
If f is an
function, then ∫−𝑎 𝑓(𝑥)𝑑𝑥 = 0
𝑎
𝜋
2
𝜋
−
2
Example 7 – Evaluate ∫ (𝑠𝑖𝑛3 𝑥 𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥)𝑑𝑥
What you should learn
How to evaluate a
definite integral
involving an even or odd
function