I need to evaluate the integral
$$ I=\int_0^{\pi/4} \sin^4 \theta \cos^5 2\theta~d\theta$$
I am thinking of changing $2\theta$ to $u$ but then I have a problem with $\sin (u/2)$. Any help would be appreciated.
2026-03-25 16:38:35.1774456715
evalute an integral with beta function 3
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Nu substitution is necessary. Use the fact that $\cos(2\theta)=1-2\sin ^{2}\theta$. Expand the powers. Note that $\int_0^{\pi/4} \sin^{n}\theta\, d\theta$ can be valuated using integration by parts.