Prove that \int_0^\frac{\pi}{2}\cos^m x \sin^m xdx=2^{-m}\int_0^\frac{\pi}{2}\cos^m xdx

Terrence Moore 2022-04-28 Answered
Prove that 0π2cosmxsinmxdx=2m0π2cosmxdx
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Answers (1)

vanvanvan4ie
Answered 2022-04-29 Author has 12 answers
Let 2x=π2u ,then 2dx=2du, so:
2m0π2sinm2xdx=2m12π2π2sinm({π2u})du=2m0π2cosμdu
This happens because cos(x) is even, therefore cosmx is symmetric over symmetric intervals.
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