Integral dari $$$\sin{\left(2 \right)} \tanh{\left(\eta \right)}$$$

Temukan $$$\int \sin{\left(2 \right)} \tanh{\left(\eta \right)}\, d\eta$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$ dengan $$$c=\sin{\left(2 \right)}$$$ dan $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$:

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

Tulis ulang tangen hiperbolik sebagai $$$\tanh\left(\eta\right)=\frac{\sinh\left(\eta\right)}{\cosh\left(\eta\right)}$$$:

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

Misalkan $$$u=\cosh{\left(\eta \right)}$$$.

Kemudian $$$du=\left(\cosh{\left(\eta \right)}\right)^{\prime }d\eta = \sinh{\left(\eta \right)} d\eta$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\sinh{\left(\eta \right)} d\eta = du$$$.

Integral tersebut dapat ditulis ulang sebagai

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

Integral dari $$$\frac{1}{u}$$$ adalah $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Ingat bahwa $$$u=\cosh{\left(\eta \right)}$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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