一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
$$
I = \int \frac{\sin ^{2} x}{\left( x \cos x – \sin x \right) ^{2}} \mathrm{~d} x
$$
难度评级:
二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
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:
$$
\begin{aligned}
I \\ \\
& = \frac{\sin x}{x} \times \frac{1}{x \cos x – \sin x} – \int \frac{1}{x ^{2}} \mathrm{~d} x \\ \\
& = \frac{\sin x}{x} \times \frac{1}{x \cos x – \sin x} + \frac{1}{x} + C \\ \\
& = \frac{1}{x} \left( \frac{\tan x}{x – \tan x} + 1 \right) + C \\ \\
& = \frac{1}{x} \left( \frac{\tan x + x – \tan x}{x – \tan x} \right) + C \\ \\
& = \textcolor{springgreen}{\boldsymbol{ \frac{1}{x – \tan x} + C }}
\end{aligned}
$$
高等数学![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
涵盖高等数学基础概念、解题技巧等内容,图文并茂,计算过程清晰严谨。
线性代数![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
以独特的视角解析线性代数,让繁复的知识变得直观明了。
特别专题![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
通过专题的形式对数学知识结构做必要的补充,使所学知识更加连贯坚实。