∫0π2cos⁡nxdx=∫0π2sin⁡nxdx={(n−1)!!(n)!!⋅π2,n为偶数;(n−1)!!(n)!!,n为奇数;={(n−1n)⋅(n−3n−2)⋯(34)⋅(12)⋅(π2),n为偶数(n−1n)⋅(n−3n−2)⋯(45)⋅(23)⋅(1),n为奇数\begin{split} &\int_0^{\frac{\pi}{2}}\cos^nxdx = \int_0^{\frac{\pi}{2}}\sin^nxdx \\ & = \begin{cases} \frac{(n-1)!!}{(n)!!}\cdot\frac{\pi}{2},& n为偶数; \\ \frac{(n-1)!!}{(n)!!},& n为奇数; \end{cases} \\ & = \begin{cases} (\frac{n-1}{n})\cdot(\frac{n-3}{n-2})\cdots(\frac{3}{4})\cdot(\frac{1}{2})\cdot(\frac{\pi}{2}),& n为偶数 \\ (\frac{n-1}{n})\cdot(\frac{n-3}{n-2})\cdots(\frac{4}{5})\cdot(\frac{2}{3})\cdot(1),& n为奇数 \end{cases} \end{split}

∫0π2sin⁡4xcos⁡2xdx=∫0π2sin⁡4x(1−sin⁡2x)dx=∫0π2sin⁡4xdx−∫0π2sin⁡6xdx=(34⋅12⋅π2)−(56⋅34⋅12⋅π2)=132π\begin{split} &\int_{0}^{\frac{\pi}{2}}\sin^4x\cos^2xdx \\ &= \int_{0}^{\frac{\pi}{2}}\sin^4x(1-\sin^2x)dx \\ &= \int_{0}^{\frac{\pi}{2}}\sin^4xdx-\int_{0}^{\frac{\pi}{2}}\sin^6xdx \\ &= (\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}) - (\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}) \\ &=\frac{1}{32}\pi \end{split}

∫0π2sin⁡4xcos⁡2xdx=12⋅B(32,52)=12⋅Γ(32)⋅Γ(52)Γ(4)=12⋅(12⋅π)⋅(32⋅12⋅π)3⋅2⋅1=π32\begin{split} &\int_{0}^{\frac{\pi}{2}}\sin^4x\cos^2xdx \\ & = \frac{1}{2}\cdot B(\frac{3}{2},\frac{5}{2}) \\ & = \frac{1}{2}\cdot\frac{\Gamma(\frac{3}{2})\cdot\Gamma(\frac{5}{2})}{\Gamma(4)} \\ & = \frac{1}{2}\cdot\frac{(\frac{1}{2}\cdot\sqrt{\pi})\cdot(\frac{3}{2}\cdot\frac{1}{2}\cdot\sqrt{\pi})}{3\cdot2\cdot1} \\ & = \frac{\pi}{32} \end{split}