In this explainer, we will learn how to evaluate integrals of functions in the form and .
Integration and differentiation are the reverse processes of each other. This means that knowing how to perform differentiation on certain expressions can help solve certain types of integration problems. Furthermore, we can verify our answer to an integration problem by differentiating it. In this sense, a very useful application of this property is that any derivative result can be stated as an integration result by reversing the process.
For example, we recall that the chain rule tells us that if is differentiable at and is differentiable at , then
We can reverse this process by integrating both sides of this result with respect to . We note that integrating a derivative leaves the function unchanged:
This is now in the form of an integral result, where we need to add a constant of integration as usual:
This is known as the reverse chain rule since it is found by reversing the chain rule by integration.
Theorem: Reverse Chain Rule
If is differentiable at and is differentiable at , then
There are many applications of the chain rule; however, in this explainer, we will focus on two specific applications of this result.
Let for some unknown constant and let be a differentiable function. We can recall that the power rule for differentiation tells us that . Substituting these expressions into the reverse chain rule, we obtain
We want to take out the factor of from the integral and divide both sides of the equation by ; however, we can only do this if :
Multiplying the constant of integration by a constant yields a constant, so we have
We can generalize this by multiplying both sides of the equation by a constant :
We have the following result.
Property: Application of the Reverse Chain Rule
For any differentiable function and real constants and , where , we have
This allows us to evaluate any integral in this form. For example, consider
We could evaluate this integral by expanding the brackets using the binomial expansion formula; however, it is easier to set in the reverse chain rule formula. We then have , and we can note that .
Therefore, if we substitute , , and into the reverse chain rule formula, we obtain
In our first example, we will apply the reverse chain rule to integrate a rational function.
Example 1: Using the Reverse Chain Rule to Integrate a Function
Determine .
Answer
In order to answer this question, we first note that we are asked to integrate the quotient of two polynomials. We can start by checking for patterns in the integrand first to see if this is in a standard form for integration. One thing we can always check is whether we can apply the reverse chain rule, which states that for any constant and differentiable function ,
To check if we can apply this result, we need to differentiate the denominator in the integrand. We call this function and differentiate it term by term using the power rule for differentiation:
We see that this is exactly equal to the numerator of the integrand, so we have
Let’s verify our answer by differentiation. We recall that for a differentiable function ,
Therefore,
This is equal to the integrand we are given, so it is the correct antiderivative. This verifies that the answer is correct.
Hence,
In our next example, we will apply the reverse chain rule to integrate the quotient of two functions.
Example 2: Using the Reverse Chain Rule to Integrate a Function
Determine .
Answer
In order to answer this question, we first note that we are asked to integrate the quotient of two functions. Integrating the quotient of functions is difficult, so we should always check for patterns in the integrand first to look for the simplest method of integration.
Let’s check if we can apply the reverse chain rule, which states that for any constant and differentiable function ,
To apply this result, we need to differentiate the function in the denominator of the integrand and compare it to the function in the numerator of the integrand. We can do this by calling the function in the denominator of the integrand and recalling that . Therefore,
This is not equal to the numerator of the integrand. However, it is a constant multiple of the numerator of the integrand:
We can use this to apply the reverse chain rule to evaluate the integral. We first take the factor of out of the integral:
Next, we can set ; then, . We rewrite as to get
This is now in the form of the reverse chain rule with , so we have
We can substitute into the result, and we note that is just a constant, so is also a constant. We can just call the final constant :
A second application of the reverse chain rule is to let , so we have . From the reverse chain rule, we have
Rearranging and multiplying through by a constant yields
We have shown the following two applications of the reverse chain rule.
Property: Application of the Reverse Chain Rule
For any differentiable function and real constant , we have
In our next example, we will apply the reverse chain rule to integrate the product of trigonometric functions.
Example 3: Using the Reverse Chain Rule to Integrate a Function
Use the reverse chain rule to find an expression for .
Answer
Let’s begin by recalling the version of the reverse chain rule that is most relevant to our integral: for a differentiable function and constants and , with ,
We can set , since this is the function raised to a power. We then have . We can then rewrite our integral in this form:
Therefore, we set , , and and apply the reverse chain rule to get
In our next example, we will use the reverse chain rule to evaluate a definite integral.
Example 4: Using the Reverse Chain Rule to Evaluate an Integral of a Function
Use the reverse chain rule to evaluate .
Answer
Let’s begin by recalling the version of the reverse chain rule that is most relevant to our integral: for a differentiable function and constants and , with ,
We know that taking the square root is the same as raising the expression to a power of , so we have
To apply the reverse chain rule, we need to set , and since this is the term raised to a power, we can differentiate term by term by using the power rule for differentiation to get
We want to compare this to ; we can see that
Therefore, we can rewrite the integral as
This is in the form of the reverse chain rule with , , and . Hence, by applying the reverse chain rule, we have
We calculate that and take this factor out of the evaluation and simplify to get
We now need to evaluate at the limits of integration:
Hence,
In our final example, we will use the reverse chain rule to evaluate the definite integral of a function in order to determine the upper bound of the integral.
Example 5: Using the Reverse Chain Rule to Find the Value of a Constant in a Definite Integral
Given that and , find the value of .
Answer
To find the value of , we first need to find an expression for in terms of . We can do this by noting that the integrand is the product of two functions, so we can try applying the reverse chain rule, which states that for a differentiable function and constants and , with ,
We can set , since this is the function that is raised to a power. We then recall that , which appears in the integral. This allows us to rewrite the integral as
This is now in the form of the reverse chain rule with , , and . Therefore,
We can now evaluate this at the limits of integration to obtain
Hence,
We are given that , so our expression for this integral must be equal to this value:
Adding 1 to both sides yields
Taking the cube root of both sides of the equation gives us
Taking the reciprocal of both sides of the equation and recalling that gives
We are given that , and there is only one solution in this interval for the equation; that is,
Hence,
Let’s finish by recapping some of the important points from this explainer.
Key Points
- The reverse chain rule states that for differentiable functions and ,
- Applying the reverse chain rule with and yields
- Applying the reverse chain rule with yields
