Does anyone have a reference which gives the explicit expression of the following integral
$$I_{m, n}=\int_{0}^{\pi} \cos^{m}(x)\, \sin^{n}(x) \, dx,$$ for any positive integers $m,n.$
Thank you you in advance?
2026-03-27 07:35:36.1774596936
Calculate $I_{m, n}=\int_{0}^{\pi} \cos^{m}(x)\, \sin^{n}(x) \, dx,$?
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As Lord Shark the Unknown noted, for even $m$ we have $$I_{m,\,n}=2\int_0^{\pi/2}\cos^m x\sin^n x\,dx=\operatorname{B}\bigg(\frac{m+1}{2},\,\frac{n+1}{2}\bigg).$$