$$ \int_{0}^{\frac{\pi}{2}} \sin ^{n}x , dx=\int_{0}^{\frac{\pi}2{}}\cos ^{n}x , dx=\left{ \begin{align} &\frac{(n-1)!!}{n!!}& (n为奇) \ &\frac{(n-1)}{n!!}\cdot \frac{\pi}{2}&(n为偶) \end{align} \right. $$ $$ \int_{0}^{\pi} \sin ^{n} x , dx = 2\cdot \int_{0}^{\frac{\pi}{2}} \sin ^{n}x , dx $$ $$ \int_{0}^{\pi} \cos ^{n}x , dx =\left{ \begin{align} & 0 & (n为奇) \ & 2\cdot \int_{0}^{\frac{\pi}{2}} \cos ^{n}x , dx & (n为偶) \end{align} \right. $$ $$ \int_{0}^{2\pi} \sin ^{n}x , dx =\int_{0}^{2\pi} \cos ^{n}x , dx=\left{ \begin{align} & 0 & (n为奇) \ & 4\cdot \int_{0}^{\frac{\pi}{2}} \sin ^{n} , dx & (n为偶) \end{align} \right.
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