Integrating Powers and Product of Sines and Cosines

These are integrals of the following form:

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We have two cases: both m and n are even or at least one of them is odd.

Case I: m or n odd


Suppose n is odd. Hence n = 2k + 1. So tex2html_wrap_inline42 hold. Therefore, we have

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which suggests the substitution tex2html_wrap_inline46 . Indeed, we have tex2html_wrap_inline48 and hence

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The latest integral is a polynomial function of u which is easy to integrate.

Remark. Note that if m is odd, then we will split tex2html_wrap_inline56 and carry the same calculations. In this case, the substitution will be tex2html_wrap_inline58 .

Example 1

Case II: m and n are even


The main idea behind is to use the trigonometric identities

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Example 2

Remark. The following two formulas may be helpful in integrating powers of sine and cosine.

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More Examples

More Challenging Problems

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Author: Mohamed Amine Khamsi

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