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 halfangle trigonometry identities:
Here’s how you integrate cos^{2} x:

Use the halfangle identity for cosine to rewrite the integral in terms of cos 2x:

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 halfangle identities to rewrite the integral in terms of cos 2x:

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 oddpowered 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: