$$$\tanh{\left(x \right)}$$$ 的積分

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

將雙曲正切改寫為 $$$\tanh\left(x\right)=\frac{\sinh\left(x\right)}{\cosh\left(x\right)}$$$:

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

令 $$$u=\cosh{\left(x \right)}$$$。

則 $$$du=\left(\cosh{\left(x \right)}\right)^{\prime }dx = \sinh{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\sinh{\left(x \right)} dx = du$$$。

該積分可改寫為

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

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

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

回顧一下 $$$u=\cosh{\left(x \right)}$$$：

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

因此，

$$\int{\tanh{\left(x \right)} d x} = \ln{\left(\cosh{\left(x \right)} \right)}$$

加上積分常數：

$$\int{\tanh{\left(x \right)} d x} = \ln{\left(\cosh{\left(x \right)} \right)}+C$$

$$$\int \tanh{\left(x \right)}\, dx = \ln\left(\cosh{\left(x \right)}\right) + C$$$A