定积分

1. $\int_0^{\frac{\pi}{2}} f(\sin x)\mathrm{d}x = \int_0^{\frac{\pi}{2}} f(\cos x) \mathrm{d}x$
2. $\int_0^\pi xf(\sin x)\mathrm{d}x = \frac{\pi}{2}\int_0^\pi f(\sin x) \mathrm{d}x$

   e.g.
   $\int_0^\pi \frac{x\sin x}{1+cos^2x} \mathrm{d}x = \frac{\pi}{2}\int_0^\pi \frac{\sin x}{2-sin^2x} \mathrm{d}x$

3. Walls公式：
   $I_n=\int_0^\frac{\pi}{2}\sin^nx\mathrm{d}x\=\begin{cases}\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ \cdots\ \cdot \frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}\quad (\text{n is even}) \\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ \cdots\ \cdot \frac{4}{5}\cdot\frac{2}{3}\quad (\text{n is odd, n>1})\end{cases}$