5.5: Substitution

Learning Objectives

• 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 . For example, in the integral

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 and part of the integrand with . It is also referred to as change of variables because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.

Substitution with Indefinite Integrals

Let , where is continuous over an interval, let be continuous over the corresponding range of , and let be an antiderivative of Then,

Proof

Let , , , and be as specified in the theorem. Then

Integrating both sides with respect to , we see that

If we now substitute , and , we get

□

Returning to the problem we looked at originally, we let and then .

Rewrite the integral (Equation ) in terms of :

Using the power rule for integrals, we have

Substitute the original expression for back into the solution:

We can generalize the procedure in the following Problem-Solving Strategy.

Problem-Solving Strategy: Integration by Substitution

1. Look carefully at the integrand and select an expression within the integrand to set equal to u. Let’s select . such that is also part of the integrand.
2. Substitute and into the integral.
3. 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 as .
4. Evaluate the integral in terms of .
5. Write the result in terms of and the expression

Example : Using Substitution to Find an Antiderivative

Use substitution to find the antiderivative of

Solution

The first step is to choose an expression for . We choose because then and we already have in the integrand. Write the integral in terms of :

Remember that is the derivative of the expression chosen for , regardless of what is inside the integrand. Now we can evaluate the integral with respect to :

Analysis

We can check our answer by taking the derivative of the result of integration. We should obtain the integrand. Picking a value for of , we let We have

so

This is exactly the expression we started with inside the integrand.

Exercise

Use substitution to find the antiderivative of

Hint

Let

Answer

Sometimes we need to adjust the constants in our integral if they don’t match up exactly with the expressions we are substituting.

Example : Using Substitution with Alteration

Use substitution to find the antiderivative of

Solution

Rewrite the integral as Let and Now we have a problem because and the original expression has only We have to alter our expression for or the integral in will be twice as large as it should be. If we multiply both sides of the equation by . we can solve this problem. Thus,

Write the integral in terms of , but pull the outside the integration symbol:

Integrate the expression in :

Exercise

Use substitution to find the antiderivative of

Hint

Multiply the du equation by .

Answer

Example : Using Substitution with Integrals of Trigonometric Functions

Use substitution to evaluate the integral

Solution

We know the derivative of is , so we set . Then

Substituting into the integral, we have

Evaluating the integral, we get

Putting the answer back in terms of t, we get

Exercise

Use substitution to evaluate the integral

Hint

Use the process from Example to solve the problem.

Answer

Exercise

Use substitution to evaluate the indefinite integral

Hint

Use the process from Example to solve the problem.

Answer

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, should be the only variable in the integrand. In some cases, this means solving for the original variable in terms of . This technique should become clear in the next example.

Example : Finding an Antiderivative Using -Substitution

Use substitution to find the antiderivative of

Solution

If we let then . But this does not account for the in the numerator of the integrand. We need to express in terms of If , then Now we can rewrite the integral in terms of

Then we integrate in the usual way, replace with the original expression, and factor and simplify the result. Thus,

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.

Substitution with Definite Integrals

Let and let be continuous over an interval , and let be continuous over the range of Then,

Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if is an antiderivative of we have

Then

and we have the desired result.

Example : Using Substitution to Evaluate a Definite Integral

Use substitution to evaluate

Solution

Let , so . Since the original function includes one factor of and , multiply both sides of the equation by Then,

To adjust the limits of integration, note that when and when

Then

Evaluating this expression, we get

Exercise

Use substitution to evaluate the definite integral

Hint

Use the steps from Example to solve the problem.

Answer

Exercise

Use substitution to evaluate

Hint

Use the process from Example to solve the problem.

Answer

Example : Using Substitution with an Exponential Function

Use substitution to evaluate

Solution

Let Then, To adjust the limits of integration, we note that when , and when . So our substitution gives

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 after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in Example .

Example : Using Substitution to Evaluate a Trigonometric Integral

Use substitution to evaluate

Solution

Let us first use a trigonometric identity to rewrite the integral. The trig identity allows us to rewrite the integral as

Then,

We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let Then, or . Also, when and when Expressing the second integral in terms of , we have

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 and for appropriate expressions in the integrand.
• When using substitution for a definite integral, we also have to change the limits of integration.

Key Equations

• Substitution with Indefinite Integrals
• Substitution with Definite Integrals

Glossary

change of variables

the substitution of a variable, such as , for an expression in the integrand

integration by substitution

a technique for integration that allows integration of functions that are the result of a chain-rule derivative