In this explainer, we will learn how to evaluate integrals of the products of trigonometric terms using trigonometric identities.
There are many trigonometric functions we cannot easily integrate. For example, imagine we were asked to determine . This is not a standard trigonometric function whose integral we know, so we would need to find a function whose derivative is . This is quite a difficult task, so instead, we can try rewriting the integrand into an easier form to integrate.
We can recall two identities involving the square of the sine function. We have the Pythagorean identity, which says that for any angle ,
If we used this identity to rewrite the integrand, we would need to integrate the square of the cosine function and this is not any easier.
Instead, we can recall the double-angle formula for the cosine function, which tells us that for any angle ,
We can rearrange this identity to find an expression for as follows:
We can integrate the left-hand side of this equation, so this will allow us to integrate the square of the sine function. We have
We then recall that and for any nonzero constant . Thus, where we combine the constants of integration into a single constant called .
Hence,
There are many different identities we can use to rewrite integrands into a form that we may be able to integrate. These include the Pythagorean identity, angle sum and difference identities, double-angle identities, and half-angle identities.
Let’s now see an example of integrating the square of the tangent function by rewriting it using the Pythagorean identity.
Example 1: Integrating Tangent Squared
Determine .
Answer
It is not immediately obvious how to directly integrate the square of the tangent function; however, we can manipulate the integrand into a form that we can integrate more easily.
We can recall that we can find an expression for the square of the tangent function by using the Pythagorean identity, which tells us that for any angle , we have
Dividing both sides of the identity by and using the fact that , we get
We can rearrange this identity to make the subject:
We now rewrite this identity by substituting to get
We can now substitute this expression into our integral. This gives us
We can take the constant factor of 3 outside of the integral to get
We can now integrate this term by term by recalling that for any nonzero constant and .
Instead of adding separate constants of integration, we can add a single constant of integration at the end of the expression to get
We can leave our answer like this; however, we can rewrite this by taking out a factor of to get
Hence, we have shown that
In our next example, we will integrate the square of the cosine function by rewriting it using a double-angle identity.
Example 2: Integrating Cosine Squared
Determine .
Answer
It is not immediately obvious how to directly integrate the square of the cosine function; however, we can manipulate the integrand into a form that we can integrate. We might be tempted to do this by using the Pythagorean identity; however, this would give us , and this expression is no easier to integrate than what we started with.
Instead, we can recall that the double-angle identity for the cosine states that for any value of , we have that
We can rearrange this identity to make the subject. We have
We can substitute this into the integral to get
We can also take out the constant factor of 4 to simplify the integral as shown:
We can now integrate each term separately by recalling that for any nonzero constant and .
We now evaluate the integral and instead of adding separate constants of integration, we can add a single constant of integration at the end of the expression to get
In our next example, we will integrate a trigonometric expression by rewriting it using various trigonometric identities and integral results.
Example 3: Integrating a Trigonometric Function Using Identities
Determine .
Answer
This integral is hard to integrate directly, so let’s instead rewrite the integrand. We can start by expanding the brackets to obtain
We can integrate two of these terms using standard integral results. We recall the following:
To integrate the term , we can rewrite it using the Pythagorean identity. We have for any angle ,
We can use this to rewrite the integral as
We can now integrate each term separately. We have
Therefore, we can add these results together (with a single constant of integration) to evaluate the integral. We have
Hence,
In the previous example, we can simplify our answer by multiplying through by . is still a constant, so we can just call this new constant . We get
In our next example, we will integrate a trigonometric function by rewriting it using trigonometric identities.
Example 4: Integrating a Trigonometric Function Using Identities
Determine .
Answer
It is hard to evaluate this integral directly, so let’s instead rewrite the integrand. We can start by multiplying out the parentheses to get
We can now divide each term in the numerator by the denominator separately, giving us
We can then rewrite the integrand using trigonometric and reciprocal trigonometric identities:
We cannot integrate each term in the integrand directly, we will rewrite using the Pythagorean identity. We recall that for any angle , we have
Substituting this into the integrand and simplifying gives us
We can now integrate each term separately by recalling the following integral results:
We can now integrate each term separately, adding a single constant of integration to get
Hence,
In our final example, we evaluate a definite integral by rewriting it using a half-angle identity.
Example 5: Evaluating a Definite Integral of a Trigonometric Function Using Identities
Evaluate .
Answer
There are many different ways we could rewrite the integrand in order to evaluate this integral. The easiest way is to note the similarity between the integrand and the half-angle identity for the tangent function::
We can use this to rewrite the integrand as shown:
We now rewrite using the Pythagorean identity. We recall that for any angle , we have
Substituting into the identity gives us
Substituting this into our integrand gives us
We can now integrate this expression term by term by recalling that for any nonzero constant and .
Substituting into this integral result yields
We can use this to evaluate the integral. We have
Now, all that is left to do is to evaluate the limits of integration. We get
Let’s finish by recapping some of the important points from this explainer.
Key Points
- We can evaluate some integrals of trigonometric functions by rewriting them using the Pythagorean identity, angle sum and difference identities, double-angle identities, and half-angle identities.
- There are often many ways to rewrite the integrands, so it is a good idea to check different options to see which gives an easy-to-evaluate integral.
