殊途同归：用两种不同的分部积分方法计算同一道题

一、题目

难度评级：

二、解析

解法一

$$
\begin{aligned}
I \\ \\
& = \int \frac{\sin ^{2} x}{\left( x \cos x – \sin x \right) ^{2}} \mathrm{~d} x \\ \\
& = \int \frac{\frac{\sin ^{2} x}{\cos ^{2} x}}{\left( \frac{x \cos x – \sin x}{\cos x} \right) ^{2}} \mathrm{~d} x \\ \\
& = \int \frac{\frac{\sin ^{2} x + \cos ^{2} x – \cos ^{2} x}{\cos ^{2} x}}{\left( x – \tan x \right) ^{2}} \mathrm{~d} x \\ \\
& = \int \frac{\frac{1}{\cos ^{2} x} – 1}{\left( x – \tan x \right) ^{2}} \mathrm{~d} x
\end{aligned}
$$

又因为：

$$
\textcolor{pink}{
\left( \frac{1}{x – \tan x} \right) ^{\prime} = \frac{- (1 – \frac{1}{\cos ^{2} x})}{(x – \tan x) ^{2}} = \frac{\frac{1}{\cos ^{2} x} – 1}{(x – \tan x) ^{2}}
}
$$

所以：

$$
\begin{aligned}
I \\ \\
& = \int \frac{\frac{1}{\cos ^{2} x} – 1}{\left( x – \tan x \right) ^{2}} \mathrm{~d} x \\ \\
& = \int \mathrm{~d} \left( \frac{1}{x – \tan x} \right) \\ \\
& = \textcolor{springgreen}{\boldsymbol{ \frac{1}{x – \tan x} + C }}
\end{aligned}
$$

解法二

由于：

$$
\textcolor{pink}{
\begin{aligned}
\left( \frac{1}{x \cos x – \sin x} \right) ^{\prime} \\ \\
& = \frac{-(\cos x – x \sin x – \cos x)}{(x \cos x – \sin x) ^{2}} \\ \\
& = \frac{x \sin x}{(x \cos x – \sin x) ^{2}}
\end{aligned}
}
$$

所以：

$$
\begin{aligned}
I \\ \\
& = \int \frac{\sin ^{2} x}{\left( x \cos x – \sin x \right) ^{2}} \mathrm{~d} x \\ \\
& = \int \frac{\sin x}{x} \mathrm{~d} \left( \frac{1}{x \cos x – \sin x} \right) \\ \\
& = \frac{\sin x}{x} \times \frac{1}{x \cos x – \sin x} – \int \frac{1}{x \cos x – \sin x} \mathrm{~d} \left( \frac{\sin x}{x} \right)
\end{aligned}
$$

又因为：

$$
\textcolor{pink}{
\left( \frac{\sin x}{x} \right) ^{\prime} = \frac{x \cos x – \sin x}{x ^{2}}
}
$$

所以：

$$
\begin{aligned}
I \\ \\
& = \frac{\sin x}{x} \times \frac{1}{x \cos x – \sin x} – \int \frac{1}{x ^{2}} \mathrm{~d} x \\ \\
& = \frac{\sin x}{x} \times \frac{1}{x \cos x – \sin x} + \frac{1}{x} + C \\ \\
& = \frac{1}{x} \left( \frac{\tan x}{x – \tan x} + 1 \right) + C \\ \\
& = \frac{1}{x} \left( \frac{\tan x + x – \tan x}{x – \tan x} \right) + C \\ \\
& = \textcolor{springgreen}{\boldsymbol{ \frac{1}{x – \tan x} + C }}
\end{aligned}
$$

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