证明定积分公式：

$$
\begin{gathered}
I_n=\int_0^{\frac{\pi}{2}} \sin ^n x d x\left(=\int_0^{\frac{\pi}{2}} \cos ^n x d x\right) \\
=\left\{\begin{array}{l}
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, n \text { 为正偶数, } \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, n \text { 为大于 } 1 \text { 的正奇数. }
\end{array}\right.
\end{gathered}
$$