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

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

定数倍の法則 $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$ を、$$$c=\frac{\cos{\left(2 \right)}}{2}$$$ と $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$ に対して適用する:

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

双曲線正接を$$$\tanh\left(\eta\right)=\frac{\sinh\left(\eta\right)}{\cosh\left(\eta\right)}$$$で表せ:

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

$$$u=\cosh{\left(\eta \right)}$$$ とする。

すると $$$du=\left(\cosh{\left(\eta \right)}\right)^{\prime }d\eta = \sinh{\left(\eta \right)} d\eta$$$（手順は»で確認できます）、$$$\sinh{\left(\eta \right)} d\eta = du$$$ となります。

この積分は次のように書き換えられる

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

$$$\frac{1}{u}$$$ の不定積分は $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$ です:

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

次のことを思い出してください $$$u=\cosh{\left(\eta \right)}$$$:

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

したがって、

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

積分定数を加える:

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

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