Integrate When the Powers of Sine, Cosine Are Even, Nonnegative

By Mark Ryan

When the powers of both sine and cosine are even and nonnegative, you can convert the integrand into odd powers of cosines by using the following trig identities.

Two handy trig identities:

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Then you finish the problem by converting the remaining cosines to sines with the Pythagorean identity, simplifying, and then integrating with substitution. Here’s an example:

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The first in this string of integrals is a no-brainer; the second is a simple reverse rule with a little tweak for the 2; you do the third integral by using the cos2 (x) identity a second time; and you handle the fourth integral as you do when the power of cosine is odd and positive. Your final answer should be

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A veritable cake walk.

Dont forget your trig identities. If you get a problem where the powers of sine and cosine aren’t both even and non-negative, try using a trig identity like

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to convert the integral into one you can handle.

For example, in

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you can use the Pythagorean identity to convert it to

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This splits up into

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and the rest is easy. Try it. See whether you can differentiate your result and arrive back at the original problem.