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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 sin2 x and cos2 x, you would use these two half-angle trigonometry identities:

image0.png

Here’s how you integrate cos2 x:

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

    image1.png
  2. Use the Constant Multiple Rule to move the denominator outside the integral:

    image2.png
  3. Distribute the function and use the Sum Rule to split it into several integrals:

    image3.png
  4. Evaluate the two integrals separately:

    image4.png

As a second example, here’s how you integrate sin2 x cos4 x:

  1. Use the two half-angle identities to rewrite the integral in terms of cos 2x:

    image5.png
  2. Use the Constant Multiple Rule to move the denominators outside the integral:

    image6.png
  3. Distribute the function and use the Sum Rule to split it into several integrals:

    image7.png
  4. Evaluate the resulting odd-powered integrals:

    image8.png

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

image9.png

Now distribute the result:

image10.png
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