$$$\frac{\cos{\left(2 \right)} \tanh{\left(\eta \right)}}{2}$$$ 的積分

求$$$\int \frac{\cos{\left(2 \right)} \tanh{\left(\eta \right)}}{2}\, d\eta$$$。

套用常數倍法則 $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$，使用 $$$c=\frac{\cos{\left(2 \right)}}{2}$$$ 與 $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$：

$${\color{red}{\int{\frac{\cos{\left(2 \right)} \tanh{\left(\eta \right)}}{2} d \eta}}} = {\color{red}{\left(\frac{\cos{\left(2 \right)} \int{\tanh{\left(\eta \right)} d \eta}}{2}\right)}}$$

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

$$\frac{\cos{\left(2 \right)} {\color{red}{\int{\tanh{\left(\eta \right)} d \eta}}}}{2} = \frac{\cos{\left(2 \right)} {\color{red}{\int{\frac{\sinh{\left(\eta \right)}}{\cosh{\left(\eta \right)}} d \eta}}}}{2}$$

令 $$$u=\cosh{\left(\eta \right)}$$$。

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

該積分可改寫為

$$\frac{\cos{\left(2 \right)} {\color{red}{\int{\frac{\sinh{\left(\eta \right)}}{\cosh{\left(\eta \right)}} d \eta}}}}{2} = \frac{\cos{\left(2 \right)} {\color{red}{\int{\frac{1}{u} d u}}}}{2}$$

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

$$\frac{\cos{\left(2 \right)} {\color{red}{\int{\frac{1}{u} d u}}}}{2} = \frac{\cos{\left(2 \right)} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

回顧一下 $$$u=\cosh{\left(\eta \right)}$$$：

$$\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)} \cos{\left(2 \right)}}{2} = \frac{\ln{\left(\left|{{\color{red}{\cosh{\left(\eta \right)}}}}\right| \right)} \cos{\left(2 \right)}}{2}$$

因此，

$$\int{\frac{\cos{\left(2 \right)} \tanh{\left(\eta \right)}}{2} d \eta} = \frac{\ln{\left(\cosh{\left(\eta \right)} \right)} \cos{\left(2 \right)}}{2}$$

加上積分常數：

$$\int{\frac{\cos{\left(2 \right)} \tanh{\left(\eta \right)}}{2} d \eta} = \frac{\ln{\left(\cosh{\left(\eta \right)} \right)} \cos{\left(2 \right)}}{2}+C$$

$$$\int \frac{\cos{\left(2 \right)} \tanh{\left(\eta \right)}}{2}\, d\eta = \frac{\ln\left(\cosh{\left(\eta \right)}\right) \cos{\left(2 \right)}}{2} + C$$$A