Use a Shortcut for Integrating Compositions of Functions
How to Integrate Even Powers of Secants with Tangents
When to Use Variable Substitution with Integrals

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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Understanding What Makes a Function Integrable
Knowing When to Integrate by Parts
How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules
Setting Up Partial Fractions When You Have Distinct Factors
Knowing When to Avoid Trigonometry Substitution
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