# How to Integrate Even Powers of Sines and Cosines

You can integrate even powers of sines and cosines. For example, if you wanted to integrate sin^{2} *x* and cos^{2} *x,* you would use these two half-angle trigonometry identities:

Here’s how you integrate cos^{2} *x**:*

Use the half-angle identity for cosine to rewrite the integral in terms of cos 2

*x**:*Use the Constant Multiple Rule to move the denominator outside the integral:

Distribute the function and use the Sum Rule to split it into several integrals:

Evaluate the two integrals separately:

As a second example, here’s how you integrate sin^{2 }*x* cos^{4 }*x**:*

Use the two half-angle identities to rewrite the integral in terms of cos 2

*x**:*Use the Constant Multiple Rule to move the denominators outside the integral:

Distribute the function and use the Sum Rule to split it into several integrals:

Evaluate the resulting odd-powered integrals:

With the integration behind you, use algebra to simplify the result. To start, combine the first and third terms, and second and fifth terms:

Now distribute the result: