套路组合积分题1

Posted 2021-11-10 Harris-H

套路组合积分题1

$$I={\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx$$

考虑令$t=\pi -x$

则
$$\begin{gathered} I={\int }_0^{\pi }\frac{(\pi -t)\sin t}{1+{\cos }^{2}t}dt \\ \Rightarrow I=\pi {\int }_0^{\pi }\frac{\sin t}{1+{\cos }^{2}t}dt-I \end{gathered}$$