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

一、题目

$$
\int_{0}^{\pi} x f(\sin x) \mathrm{d} x = ?
$$

难度评级：

二、解析

$$
\int_{0}^{\pi} x f(\sin x) \mathrm{d} x \Rightarrow
$$

令 $u$ $=$ $\pi$ $-$ $x$, $x$ $=$ $\pi$ $-$ $u$, 则：

$$
– \int_{\pi}^{0} (\pi – u) f[\sin (\pi – u)] \mathrm{d} u \Rightarrow
$$

$$
\int_{0}^{\pi} (\pi – u) f[\sin (\pi – u)] \mathrm{d} u \Rightarrow
$$

Next

由于 $\sin(\pi – u)$ $=$ $\sin u$ $\Rightarrow$

$$
\int_{0}^{\pi} (\pi – u) f(\sin u) \mathrm{d} u \Rightarrow
$$

$$
\pi \int_{0}^{\pi} f(\sin u) \mathrm{d} u – u \int_{0}^{\pi} f(\sin u) \mathrm{d} u \Rightarrow
$$

Next

再令 $u$ $=$ $x$ $\Rightarrow$

用什么字母表示变量都只是一种形式，因此，可以直接将式子中所有的变量 $x$ 替换为 $u$.

$$
\pi \int_{0}^{\pi} f(\sin x) \mathrm{d} x – x \int_{0}^{\pi} f(\sin x) \mathrm{d} x \Rightarrow
$$

$$
\int_{0}^{\pi} x f(\sin x) \mathrm{d} x = \pi \int_{0}^{\pi} f(\sin x) \mathrm{d} x – x \int_{0}^{\pi} f(\sin x) \mathrm{d} x \Rightarrow
$$

$$
2 \int_{0}^{\pi} x f(\sin x) \mathrm{d} x = \pi \int_{0}^{\pi} f(\sin x) \mathrm{d} x \Rightarrow
$$

$$
\int_{0}^{\pi} x f(\sin x) \mathrm{d} x = \frac{\pi}{2} \int_{0}^{\pi} f(\sin x) \mathrm{d} x.
$$

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