5.5: Substitution
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- Use substitution to evaluate indefinite integrals.
- Use substitution to evaluate definite integrals.
The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy. In this section we examine a technique, called integration by substitution, to help us find antiderivatives. Specifically, this method helps us find antiderivatives when the integrand is the result of a chain-rule derivative.
At first, the approach to the substitution procedure may not appear very obvious. However, it is primarily a visual task—that is, the integrand shows you what to do; it is a matter of recognizing the form of the function. So, what are we supposed to see? We are looking for an integrand of the form
we have
and
Then
and
and we see that our integrand is in the correct form. The method is called substitution because we substitute part of the integrand with the variable
Let
Let
Integrating both sides with respect to
If we now substitute
□
Returning to the problem we looked at originally, we let
Rewrite the integral (Equation
Using the power rule for integrals, we have
Substitute the original expression for
We can generalize the procedure in the following Problem-Solving Strategy.
- Look carefully at the integrand and select an expression
within the integrand to set equal to u. Let’s select𝑔 ( 𝑥 ) . such that𝑔 ( 𝑥 ) is also part of the integrand.𝑔 ′ ( 𝑥 ) - Substitute
and𝑢 = 𝑔 ( 𝑥 ) into the integral.𝑑 𝑢 = 𝑔 ′ ( 𝑥 ) 𝑑 𝑥 . - We should now be able to evaluate the integral with respect to
. If the integral can’t be evaluated we need to go back and select a different expression to use as𝑢 .𝑢 - Evaluate the integral in terms of
.𝑢 - Write the result in terms of
and the expression𝑥 𝑔 ( 𝑥 ) .
Use substitution to find the antiderivative of
Solution
The first step is to choose an expression for
Remember that
Analysis
We can check our answer by taking the derivative of the result of integration. We should obtain the integrand. Picking a value for
so
This is exactly the expression we started with inside the integrand.
Use substitution to find the antiderivative of
- Hint
-
Let
𝑢 = 𝑥 3 − 3 .
- Answer
-
∫ 3 𝑥 2 ( 𝑥 3 − 3 ) 2 𝑑 𝑥 = 1 3 ( 𝑥 3 − 3 ) 3 + 𝐶
Sometimes we need to adjust the constants in our integral if they don’t match up exactly with the expressions we are substituting.
Use substitution to find the antiderivative of
Solution
Rewrite the integral as
Write the integral in terms of
Integrate the expression in
Use substitution to find the antiderivative of
- Hint
-
Multiply the du equation by
.1 3
- Answer
-
∫ 𝑥 2 ( 𝑥 3 + 5 ) 9 𝑑 𝑥 = ( 𝑥 3 + 5 ) 1 0 3 0 + 𝐶
Use substitution to evaluate the integral
Solution
We know the derivative of
Substituting into the integral, we have
Evaluating the integral, we get
Putting the answer back in terms of t, we get
Use substitution to evaluate the integral
- Hint
-
Use the process from Example
to solve the problem.5 . 5 . 3
- Answer
-
∫ c o s 𝑡 s i n 2 𝑡 𝑑 𝑡 = − 1 s i n 𝑡 + 𝐶
Use substitution to evaluate the indefinite integral
- Hint
-
Use the process from Example
to solve the problem.5 . 5 . 3
- Answer
-
∫ c o s 3 𝑡 s i n 𝑡 𝑑 𝑡 = − c o s 4 𝑡 4 + 𝐶
Sometimes we need to manipulate an integral in ways that are more complicated than just multiplying or dividing by a constant. We need to eliminate all the expressions within the integrand that are in terms of the original variable. When we are done,
Use substitution to find the antiderivative of
Solution
If we let
Then we integrate in the usual way, replace
Substitution for Definite Integrals
Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.
Let
Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if
Then
and we have the desired result.
Use substitution to evaluate
Solution
Let
To adjust the limits of integration, note that when
Then
Evaluating this expression, we get
Use substitution to evaluate the definite integral
- Hint
-
Use the steps from Example
to solve the problem.5 . 5 . 5
- Answer
-
∫ 0 − 1 𝑦 ( 2 𝑦 2 − 3 ) 5 𝑑 𝑦 = 9 1 3
Use substitution to evaluate
- Hint
-
Use the process from Example
to solve the problem.5 . 5 . 5
- Answer
-
∫ 1 0 𝑥 2 c o s ( 𝜋 2 𝑥 3 ) 𝑑 𝑥 = 2 3 𝜋 ≈ 0 . 2 1 2 2
Use substitution to evaluate
Solution
Let
Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for
Use substitution to evaluate
Solution
Let us first use a trigonometric identity to rewrite the integral. The trig identity
Then,
We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let
Key Concepts
- Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable
and𝑢 for appropriate expressions in the integrand.𝑑 𝑢 - When using substitution for a definite integral, we also have to change the limits of integration.
Key Equations
- Substitution with Indefinite Integrals
∫ 𝑓 [ 𝑔 ( 𝑥 ) ] 𝑔 ′ ( 𝑥 ) 𝑑 𝑥 = ∫ 𝑓 ( 𝑢 ) 𝑑 𝑢 = 𝐹 ( 𝑢 ) + 𝐶 = 𝐹 ( 𝑔 ( 𝑥 ) ) + 𝐶 - Substitution with Definite Integrals
∫ 𝑏 𝑎 𝑓 ( 𝑔 ( 𝑥 ) ) 𝑔 ′ ( 𝑥 ) 𝑑 𝑥 = ∫ 𝑔 ( 𝑏 ) 𝑔 ( 𝑎 ) 𝑓 ( 𝑢 ) 𝑑 𝑢
Glossary
- change of variables
- the substitution of a variable, such as
, for an expression in the integrand𝑢
- integration by substitution
- a technique for integration that allows integration of functions that are the result of a chain-rule derivative