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Trigonometric integrals involving powers

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I used the product to sum identities rule since the integral involved cosines of different angles. I have attached a word document with the integral to solve and my work. I want to know if my answer is correct. If my answer is not correct, I want to know the correct answer and the steps to get it. Thanks.

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The trigonometric integrals involving powers are given.

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The integral to evaluate:

My Answer:

My Work:

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