定理 Wallis 积分定理

$\forall n \in \mathbb{N}$

成立

$$ \bbox[5pt]{ \int_{0}^{\frac{\pi}{2}}{\sin^n{x}}{dx} = \int_{0}^{\frac{\pi}{2}}{\cos^n{x}}{dx} = \left\{\begin{aligned} & \frac{(n-1)!!}{n!!} & n\text{为奇数}\\ & \frac{(n-1)!!}{n!!}\times\frac{\pi}{2} & n\text{为偶数} \end{aligned}\right. } $$

证明