解决三角函数定积分的组合拳：区间再现与点火公式

一、题目

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

难度评级：

二、解析

首先，利用区间在线公式，令 $t$ $=$ $\frac{\pi}{2}$ $+$ $0$ $-$ $x$, 则：

$$
t = \frac{\pi}{2} – x
$$

$$
x = \frac{\pi}{2} – t
$$

$$
\mathrm{d} x = – \mathrm{d} t
$$

$$
x \in (0, \frac{\pi}{2}) \Rightarrow t \in (\frac{\pi}{2}, 0)
$$

Next

于是：

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

$$
(-1) \int_{\frac{\pi}{2}}^{0} (\frac{\pi}{2} – t) \sin^{2} (\frac{\pi}{2} – t) \cos^{2} (\frac{\pi}{2} – t) \mathrm{d} t =
$$

$$
\int_{0}^{\frac{\pi}{2}} (\frac{\pi}{2} – t) \sin (\frac{\pi}{2} – t) \sin (\frac{\pi}{2} – t) \cos (\frac{\pi}{2} – t) \cos (\frac{\pi}{2} – t) \mathrm{d} t \Rightarrow
$$

Next

三角函数诱导公式（奇变偶不变，符号看象限） $\Rightarrow$

$$
\int_{0}^{\frac{\pi}{2}} (\frac{\pi}{2} – t) \sin t \sin t \cos t \cos t \mathrm{d} t =
$$

$$
\int_{0}^{\frac{\pi}{2}} (\frac{\pi}{2} – t) \sin^{2} t \cos^{2} t \mathrm{d} t =
$$

$$
\frac{\pi}{2} \int_{0}^{\frac{\pi}{2}} \sin^{2} t \cos^{2} t \mathrm{d} t – \int_{0}^{\frac{\pi}{2}} t \sin^{2} t \cos^{2} t \mathrm{d} t \Rightarrow
$$

Next

令 $t$ $=$ $x$ $\Rightarrow$

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

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

$$
\int_{0}^{\frac{\pi}{2}} x \sin^{2} x \cos^{2} x \mathrm{d} x = \frac{\pi}{4} \int_{0}^{\frac{\pi}{2}} \sin^{2} x \cos^{2} x \mathrm{d} x =
$$

$$
\frac{\pi}{4} \int_{0}^{\frac{\pi}{2}} \sin^{2} x (1 – \sin^{2} x) \mathrm{d} x
$$

$$
\frac{\pi}{4} \Bigg[ \int_{0}^{\frac{\pi}{2}} \sin^{2} x \mathrm{d} x – \int_{0}^{\frac{\pi}{2}} \sin^{4} x \mathrm{d} x \Bigg] =
$$

Next

点火公式 $\Rightarrow$

$$
\frac{\pi}{4} (\frac{1}{2} \cdot \frac{\pi}{2} – \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}) =
$$

$$
\frac{\pi}{4} (1 \cdot \frac{1}{2} \cdot \frac{\pi}{2} – \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}) =
$$

$$
\frac{\pi}{4} \cdot \frac{1}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi^{2}}{64}.
$$

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