$$$\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}$$$ 的積分

求$$$\int \frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$。

簡化被積函數:

$${\color{red}{\int{\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{\left(-1 + \frac{1}{\sin^{2}{\left(x \right)}}\right)d x}}}$$

逐項積分：

$${\color{red}{\int{\left(-1 + \frac{1}{\sin^{2}{\left(x \right)}}\right)d x}}} = {\color{red}{\left(- \int{1 d x} + \int{\frac{1}{\sin^{2}{\left(x \right)}} d x}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$\int{\frac{1}{\sin^{2}{\left(x \right)}} d x} - {\color{red}{\int{1 d x}}} = \int{\frac{1}{\sin^{2}{\left(x \right)}} d x} - {\color{red}{x}}$$

將被積函數改寫為以餘割函數表示:

$$- x + {\color{red}{\int{\frac{1}{\sin^{2}{\left(x \right)}} d x}}} = - x + {\color{red}{\int{\csc^{2}{\left(x \right)} d x}}}$$

$$$\csc^{2}{\left(x \right)}$$$ 的積分是 $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$：

$$- x + {\color{red}{\int{\csc^{2}{\left(x \right)} d x}}} = - x + {\color{red}{\left(- \cot{\left(x \right)}\right)}}$$

因此，

$$\int{\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}} d x} = - x - \cot{\left(x \right)}$$

加上積分常數：

$$\int{\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}} d x} = - x - \cot{\left(x \right)}+C$$

$$$\int \frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}\, dx = \left(- x - \cot{\left(x \right)}\right) + C$$$A