定积分

有关三角函数的定积分

1. 设$f(x)$在$[0,1]$上连续，则${\displaystyle {\int }_0^{\pi /2}f(\sin x)={\int }_0^{\pi /2}f(\cos x)}$

2. $I_n={\displaystyle {\int }_0^{\pi /2}{\sin }^{n}x={\int }_0^{\pi /2}{\cos }^{n}x,I_n=\frac{n-1}{n}{I}_{n-2},I_1=1,I_0=\frac{\pi }{2}}$

3. 设$f(x)$在$[0,1]$上连续，则${\displaystyle {\int }_0^{\pi }f(\sin x)=2{\int }_0^{\pi /2}f(\sin x)}$

4. 设$f(x)$在$[0,1]$上连续，则${\displaystyle {\int }_0^{\pi }xf(\sin x)=\frac{\pi }{2}{\int }_0^{\pi /2}f(\sin x)}$