一、题目
$$
\int_{0}^{\pi} x \sin^{2} x \mathrm{d} x = ?
$$
难度评级：
二、解析 
由《这篇文章》的结论可知：
$$
\int_{0}^{\pi} x f(\sin x) \mathrm{d} x = \frac{\pi}{2} \int_{0}^{\pi} f(\sin x) \mathrm{d} x.
$$
Next 
于是：
$$
\int_{0}^{\pi} x \sin^{2} x \mathrm{d} x =
$$
$$
\frac{\pi}{2} \int_{0}^{\pi} \sin^{2} x \mathrm{d} x.
$$
Next
进而，根据区间的对称性可知：
$$
\frac{\pi}{2} \int_{0}^{\pi} \sin^{2} x \mathrm{d} x =
$$
$$
\frac{\pi}{2} \cdot 2 \cdot \int_{0}^{\frac{\pi}{2}} \sin^{2} x \mathrm{d} x =
$$
$$
\pi \cdot \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi^{2}}{4}.
$$
