In this explainer, we will learn how to use integration by substitution for indefinite integrals.
Integration by substitution, also known as “-substitution” or “change of variables”, is a method of finding unknown integrals by replacing one variable with another and changing the integrand into something that is known or can be easily integrated using other methods. After performing the integration, we usually change back to our original variable by reversing the substitution to give the final result in terms of that variable.
The ability to carry out integration by substitution is a skill that develops with practice and experience. This is why it is best to look at a lot of examples and practice as much as possible. Sometimes, an apparently sensible substitution does not lead to an integral that is easy to evaluate and you must be prepared to try an alternative change of variable.
We need to be able to write the integrand in a particular way as integration by substitution is usually applied to an integral when it takes the special form
We can use the chain rule and the fundamental theorem of calculus to derive a substitution rule for integrals of this type. We recall the first part of the fundamental theorem of calculus: if is a continuous real-valued function on some interval and (i.e., is an antiderivative of ), then we have the indefinite integral where is known as the constant of integration. Also, recall the chain rule for derivatives of composite functions: if and are differentiable functions, then the chain rule expresses the derivative of their composite as
Now, suppose is an antiderivative of and is a differentiable function. We can apply the chain rule to obtain
Therefore, from the fundamental theorem of calculus,
This leads us to the following substitution rule, which is similar to the chain rule for differentiation but in reverse.
Definition: Substitution Rule
If is a differentiable function whose range is an interval and is continuous on , then where .
The key to finding the right substitution, , is finding a piece of the integrand whose derivative is also in the integrand. It is usually chosen from the “complicated” part of the integrand that we want to simplify, which can be written as a composite function for some that is continuous in a subset of the range of .
When applying this substitution rule for indefinite integrals, we substitute , integrate with respect to the variable , and reverse the substitution in the resulting antiderivative to express the final result back in terms of the variable .
We may also manipulate differentials with derivatives by treating them as a fraction, which is useful mathematical shorthand but may not be mathematically rigorous. The reason we can do this is the chain rule or
However, for the purposes of this explainer and simplicity, we may treat the derivative as a fraction as it is relevant for problems with integration by substitution.
The best way to understand this method is to see it in action. Consider the indefinite integral
In order to evaluate this, we could expand out the integrand and use the power rule for integration, but here we will use the substitution rule. We can recognize that a part of the integrand contains a composite function: with and . Since is a polynomial, it is continuous in the range of and we can use the substitution
Taking the derivative, we have or, equivalently by manipulating the differentials,
Thus, the integral can be written as
We can now substitute back to do a reverse substitution to get the final result in terms of . Finally, we have
An integral that starts out difficult can sometimes become very easy with an appropriate substitution. Let’s consider the following more complicated-looking integral:
For this integral, we could expand using the binomial theorem then multiply the result by and integrate using the power rule for integration. However, this can be tedious and prone to errors. A better way to find the integral would be by substitution.
Before we talk about the substitution, notice what happens when we differentiate using the chain rule (or the general power rule):
This is equal to the integrand up to a constant factor. In particular, we can divide by 27 to obtain
We can integrate both sides of the expression using the fundamental theorem of calculus to obtain
This technique is known as recognition, but integration by substitution does this automatically without having to find the derivative and compare terms. We can recognize that a part of the integrand contains a composite function: with and . Since is a polynomial, it is continuous in the range of and we can use the substitution
Taking the derivative of this with respect to , we have or, equivalently after manipulating the differentials,
We can now rewrite the integral in terms of the variable using the substitution rule and integrate the result using the power rule for integration:
Finally, we substitute back to get the final result in terms of , which is the same result we get from recognition.
Most of the time when you make a substitution, you only need to replace one part of the integrand as the other part, involving , would be cancelled by the expression for . However, if this does not happen, then you may need to replace any by making it the subject in the substitution. To see this in action, consider the integral
We can recognize that a part of the integrand contains a composite function: with and . Since is a polynomial, it is continuous in the range of and we can use the substitution
Taking the derivative, we have or, equivalently by manipulating the differentials,
We can now rewrite the integral in terms of the variable using the substitution rule:
Notice that we still have an appearing in the integrand that did not get cancelled, so we have to replace this by by making the subject in the substitution:
Therefore, we can replace everything in terms of and integrate the result using the power rule for integration:
Finally, we can make the reverse substitution to rewrite the result in terms of :
Now, let’s look at a few examples in order to practice and deepen our understanding. The first two examples are about finding the indefinite integral of a function involving polynomials in .
Example 1: Finding the Integration of a Function Using Integration by Substitution
Determine .
Answer
In this example, we want to find the indefinite integral of a polynomial function using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is a polynomial, it is continuous in the range of and we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Next, we apply this substitution to the integral to change the variable from to and integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
Example 2: Finding the Integration of a Function Using Integration by Substitution
Determine by using the substitution method.
Answer
In this example, we want to find the indefinite integral of a polynomial function using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is a polynomial, it is continuous in the range of and we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Next, we apply this substitution to the integral to change the variable from to :
Notice that we still have an in the integrand that did not get cancelled, so we have to replace with by making the subject in the substitution:
We can use this to eliminate any that appears in the integrand and then integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
The next example involves finding the indefinite integral of a root function.
Example 3: Finding the Integration of a Function Involving a Root Function Using Integration by Substitution
Determine .
Answer
In this example, we want to find the indefinite integral of a function involving a root using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is continuous in a subset of the range of (excluding for ), we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Next, we apply this substitution to the integral to change the variable from to , noting that appears in the integrand. We then integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
The next example involves finding the indefinite integral of trigonometric functions.
Example 4: Finding the Integration of a Function Involving Trigonometric Functions Using Integration by Substitution
Determine .
Answer
In this example, we want to find the indefinite integral of a trigonometric function using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is a polynomial, it is continuous in the range of and we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Now, we apply this substitution to the integral to change the variable from to and integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
The next example involves finding the indefinite integral of a logarithmic function.
Example 5: Finding the Integration of a Function Involving a Logarithmic Function Using Integration by Substitution
Determine .
Answer
In this example, we want to find the indefinite integral of a function involving logarithms using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is continuous in the range of , we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Now, we apply this substitution to the integral to change the variable from to and integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
The next example involves finding the indefinite integral of reciprocal trigonometric functions.
Example 6: Integrating Reciprocal Trigonometric Functions
Determine .
Answer
In this example, we want to find the indefinite integral of a reciprocal trigonometric a function using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is continuous in a subset of the range of , we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Now, we apply this substitution to the integral to change the variable from to and integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
The last example is about finding the indefinite integral involving both exponential and trigonometric functions.
Example 7: Finding the Integration of a Function Involving Trigonometric and Exponential Functions Using Integration by Substitution
Determine .
Answer
In this example, we want to find the indefinite integral of an integrand involving an exponential and trigonometric function using integration by substitution.
We first note that a part of the integrand contains a composite function: with and . Since is an exponential function, it is continuous in the range of and we can use the substitution
The derivative of this with respect to is or, equivalently by manipulating the differentials,
Now, we apply this substitution to the integral to change the variable from to and integrate the resulting expression using the power rule for integration:
Finally, we apply the reverse substitution to get the final result in terms of :
Key Points
- Integration by substitution can be used to find the indefinite integral of complicated functions involving roots, trigonometric functions, logarithmic functions, and many more.
- The substitution rule we use is similar to the chain rule for differentiation, but in reverse:
- To choose our substitution, , we look for a factor
of the integrand whose derivative also appears in the integrand or the
inner function of the “complicated” part ot the integrand.
In particular, if part of the integrand contains a derivative and/or a composite function of the form , we use the substitution .
