$$$- \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cot{\left(x \right)}}{\sin{\left(x \right)}}$$$ 的积分

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

逐项积分：

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

改写被积函数:

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

设$$$u=\csc{\left(x \right)}$$$。

则$$$du=\left(\csc{\left(x \right)}\right)^{\prime }dx = - \cot{\left(x \right)} \csc{\left(x \right)} dx$$$ (步骤见»)，并有$$$\cot{\left(x \right)} \csc{\left(x \right)} dx = - du$$$。

因此，

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

对 $$$c=-1$$$ 和 $$$f{\left(u \right)} = 1$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$- \int{\sin{\left(x \right)} \cos{\left(x \right)} d x} + {\color{red}{\int{\left(-1\right)d u}}} = - \int{\sin{\left(x \right)} \cos{\left(x \right)} d x} + {\color{red}{\left(- \int{1 d u}\right)}}$$

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=1$$$：

$$- \int{\sin{\left(x \right)} \cos{\left(x \right)} d x} - {\color{red}{\int{1 d u}}} = - \int{\sin{\left(x \right)} \cos{\left(x \right)} d x} - {\color{red}{u}}$$

回忆一下 $$$u=\csc{\left(x \right)}$$$:

$$- \int{\sin{\left(x \right)} \cos{\left(x \right)} d x} - {\color{red}{u}} = - \int{\sin{\left(x \right)} \cos{\left(x \right)} d x} - {\color{red}{\csc{\left(x \right)}}}$$

设$$$u=\sin{\left(x \right)}$$$。

则$$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步骤见»)，并有$$$\cos{\left(x \right)} dx = du$$$。

因此，

$$- \csc{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} \cos{\left(x \right)} d x}}} = - \csc{\left(x \right)} - {\color{red}{\int{u d u}}}$$

应用幂法则 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=1$$$：

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

回忆一下 $$$u=\sin{\left(x \right)}$$$:

$$- \csc{\left(x \right)} - \frac{{\color{red}{u}}^{2}}{2} = - \csc{\left(x \right)} - \frac{{\color{red}{\sin{\left(x \right)}}}^{2}}{2}$$

因此，

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

加上积分常数：

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

$$$\int \left(- \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cot{\left(x \right)}}{\sin{\left(x \right)}}\right)\, dx = \left(- \frac{\sin^{2}{\left(x \right)}}{2} - \csc{\left(x \right)}\right) + C$$$A