$$$9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}$$$ 关于$$$x$$$的积分

求$$$\int 9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$。

对 $$$c=9 i n t$$$ 和 $$$f{\left(x \right)} = \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(9 i n t \int{\sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$

改写被积函数:

$$9 i n t {\color{red}{\int{\sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 9 i n t {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}}$$

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$$$\cos{\left(x \right)} dx = du$$$。

积分变为

$$9 i n t {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = 9 i n t {\color{red}{\int{\frac{1}{u} d u}}}$$

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$9 i n t {\color{red}{\int{\frac{1}{u} d u}}} = 9 i n t {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回忆一下 $$$u=\sin{\left(x \right)}$$$:

$$9 i n t \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 9 i n t \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)}$$

因此，

$$\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x} = 9 i n t \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

加上积分常数：

$$\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x} = 9 i n t \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$$

$$$\int 9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}\, dx = 9 i n t \ln\left(\left|{\sin{\left(x \right)}}\right|\right) + C$$$A