Section 4.5 Integration by Substitution
IB Calculus
Mrs. Francis
In this lesson you learn how to evaluate different types of
definite and indefinite integrals using a variety of methods.
I. Pattern Recognition (Pages 297ο€­299)
The role of substitution in integration is comparable to the role of
in differentiation.
= 𝑭′ (π’ˆ(𝒙))π’ˆ′(𝒙)
From the definition of an antiderivative, it follows that:
∫ 𝑭′ (π’ˆ(𝒙))π’ˆ′ (𝒙)𝒅𝒙 = 𝑭(π’ˆ(𝒙)) + π‘ͺ
Antidifferentiation of a Composite Function
∫ 𝑭′ (π’ˆ(𝒙))π’ˆ′ (𝒙)𝒅𝒙 = 𝑭(π’ˆ(𝒙)) + π‘ͺ
Letting 𝒖 = π’ˆ(𝒙) gives 𝒅𝒖 = π’ˆ′ (𝒙)𝒅𝒙 and
From the definition of an antiderivative, it follows that:
∫ 𝒇(𝒖)𝒅𝒖 = 𝑭(𝒖) + π‘ͺ
Example 1:
∫(π’™πŸ + 𝟏) (πŸπ’™)𝒅𝒙
Example 2:
∫ πŸ“ 𝒄𝒐𝒔 πŸ“π’™ 𝒅𝒙
What you should
learn How to use
pattern recognition
to find an indefinite
Many integrands contain the essential part (the variable part) of
𝑔′(π‘₯) but are missing a constant multiple. In such cases, you can
∫ π’Œ 𝒇(𝒙)𝒅𝒙 = π’Œ ∫ 𝒇(𝒙)𝒅𝒙
Example 3: ∫ 𝒙(π’™πŸ + 𝟏)𝟐 𝒅𝒙
II. Change of Variables (Pages 300ο€­301)
What you should learn
You should learn how
to integrate using
With a formal change of variables, you completely
 f (g(x))gο‚’ (x) dx 
Example 4:
a. Find ∫ √2π‘₯ − 1 𝑑π‘₯
b. Find ∫ 𝑠𝑖𝑛2 3π‘₯ cos 3π‘₯ 𝑑π‘₯
Guidelines For Making Change of Variables
1. Choose a substitution 𝑒 = 𝑔(π‘₯). Usually it is best to choose the inner
part of a composite function, such as a quantity raised to a power.
2. Compute 𝑑𝑒 = 𝑔′ (π‘₯)𝑑π‘₯
3. Rewrite the integral in terms of the variable𝑒.
4. Find the resulting integral in terms of 𝑒.
5. Replace 𝑒 by 𝑔(π‘₯) to obtain an antiderivative in terms of π‘₯.
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
∫ 3(3π‘₯ − 1)4 𝑑π‘₯
∫(2π‘₯ + 1)(π‘₯ 2 + π‘₯)𝑑π‘₯
∫ 3π‘₯ 2 √π‘₯ 3 − 2 𝑑π‘₯
∫ (1−2π‘₯ 2)2 𝑑π‘₯
∫ π‘π‘œπ‘  2 π‘₯ sin π‘₯ 𝑑π‘₯
IV. Change of Variables for Definite Integrals
(Pages 303ο€­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 ο€½ο€ 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
Evaluate ∫0 π‘₯(π‘₯ 2 + 1)3 𝑑π‘₯
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 [ο€­ο€ a, a].
If f is an
function, then ∫−π‘Ž 𝑓(π‘₯)𝑑π‘₯ = 2 ∫0 𝑓(π‘₯)𝑑π‘₯
If f is an
function, then ∫−π‘Ž 𝑓(π‘₯)𝑑π‘₯ = 0
Example 7 – Evaluate ∫ (𝑠𝑖𝑛3 π‘₯ π‘π‘œπ‘  π‘₯ + 𝑠𝑖𝑛π‘₯ π‘π‘œπ‘ π‘₯)𝑑π‘₯
What you should learn
How to evaluate a
definite integral
involving an even or odd