Integration using sin and cos identities

We sometimes need to integrate functions that involve powers of trigonometric functions, or even products of powers of different trigonometric functions. We can often use trigonometric identities to help solve these problems.

By trigonometric identities, we mean the well-known identities from geometry, such as the double-angle rule, and good old Pythagoras. Those rules aren't just for triangles, they apply to calculus, too. We will see how to do that in this article.

We will only cover sine and cosine here, other trig functions will be covered in a later article.

Trigonometric identities

These standard identities are useful for this technique:

These double-angle identities are also useful. The can be derived from the third and fourth identities above by setting β equal to α:

Finally, this identity is basically Pythagoras theorem.

We will apply these to three different situations.

Odd powers of sine or cosine

We will start by solving the following integral:

We will express cos to the power 5 like this:

We can now convert the cos squared terms to sin squared using Pythagoras:

We can now write the expression now in a very specific form - a function of sin x, multiplied by cos x:

This form allows us to perform integration by substitution, using the following values for u and du:

This works well because we have a cos x dx term, with everything else expressed in terms of sin x. We apply the substitution and then multiply out the brackets:

This is a simple polynomial in u. Here is what we get when we integrate it:

All we need to do now is replace u with sin x to get the final result:

Even powers of sine or cosine

For even powers, we take a different approach. Let's look at this example:

To solve this, we will express the integrand in terms of sin x squared, and then use the double-angle formula to get rid of the square. So we first rewrite the integrand like this:

Using the double-angle formula gives:

Now we multiply out the square term:

We are getting there, but we now have a term in cos 2x squared. We need to apply another double-angle formula, but this time in 2x, which looks like this:

Making this substitution and simplifying gives:

This is just a linear combination of cosines, so we can integrate it. Just as a reminder, the integral of cos nx is a standard result:

Applying this and simplifying gives the answer:

Combining sines and cosines with different angle multipliers

Sometimes we will see problems of the form:

These problems can involve any combination of sines and cosines. As an example, we will look at a situation involving two cosines:

We will apply the trig identity for cos α times cos β from earlier:

This is just the sum of two cosines, which we can easily integrate:

More complex expressions

We have seen three techniques for dealing with products or powers of sine and cosine functions.

For more complex problems, we might need to apply the techniques more than once (as we saw in the second example). Sometimes we might need to apply more than one of the techniques shown. But the approach is always the same, we work on the integrand until it reduces to a linear combination of simple trig functions, and then perform the integration.

Related articles

• What is integration
• Mean value theorem for integrals
• Fundamental theorem of calculus
• Improper integrals
• Integration by substitution
• Integration by parts
• Integration by parts - LIATE rule
• Finding arc length by integration
• Volume of revolution
• Gabriel's horn