计算定积分 $\int_{0}^{\pi}$ $x \sin^{2} x$ $\mathrm{d} x$

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：

$$
\frac{\pi}{2} \int_{0}^{\pi} \sin^{2} x \mathrm{d} x =
$$

$$
\frac{\pi}{2} \cdot 2 \cdot \int_{0}^{\frac{\pi}{2}} \sin^{2} x \mathrm{d} x =
$$

$$
\pi \cdot \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi^{2}}{4}.
$$

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