$$$\coth{\left(x \right)}$$$ 的积分

求$$$\int \coth{\left(x \right)}\, dx$$$。

将双曲余切改写为 $$$\coth\left(x\right)=\frac{\cosh\left(x\right)}{\sinh\left(x\right)}$$$:

$${\color{red}{\int{\coth{\left(x \right)} d x}}} = {\color{red}{\int{\frac{\cosh{\left(x \right)}}{\sinh{\left(x \right)}} d x}}}$$

设$$$u=\sinh{\left(x \right)}$$$。

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

所以，

$${\color{red}{\int{\frac{\cosh{\left(x \right)}}{\sinh{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

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

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

回忆一下 $$$u=\sinh{\left(x \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sinh{\left(x \right)}}}}\right| \right)}$$

因此，

$$\int{\coth{\left(x \right)} d x} = \ln{\left(\left|{\sinh{\left(x \right)}}\right| \right)}$$

加上积分常数：

$$\int{\coth{\left(x \right)} d x} = \ln{\left(\left|{\sinh{\left(x \right)}}\right| \right)}+C$$

$$$\int \coth{\left(x \right)}\, dx = \ln\left(\left|{\sinh{\left(x \right)}}\right|\right) + C$$$A