# Powers of Sine and Cosine

Jun 21st, 2015
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Powers of Sine and CosineWe will give a general method to solve generally integrands of the form . First let us work through an example.Notice that the integrand contains an odd power of cos. So rewrite it asWe can solve this by making the substitution so . Then we can write the whole integrand in terms of by using the identity.SoThis method works whenever there is an odd power of sine or cosine.To evaluate when either or is odd.If is odd substitute and use the identity .If is odd substitute and use the identity .ExampleFind .As there is an odd power of we let so . Notice that when we have and when we have .When both and are even things get a little more complicated.To evaluate when both and are even.Use the identities and .ExampleFind As and we haveand expanding, the integrand becomesUsing the multiple angle identitiesthen we obtain on evaluatingPowers of Tan and SecantTo evaluate .1. If is even and then substitute and use the identity .2. If and are both odd then substitute and use the identity .3. If is odd and is even then use the identity and apply a reduction formula to integrate , using the examples below to integrate when .Example 1Find .There is an even power of . Substituting gives soExample 2Find .Let so . ThenExample 3Find .The trick to do this is to multiply and divide by the same thing like this:Making the substitution so More trigonometric combinationsFor the

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