Integral de $$$\cos{\left(2 \right)} \tanh{\left(\eta \right)}$$$

Encontre $$$\int \cos{\left(2 \right)} \tanh{\left(\eta \right)}\, d\eta$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$ usando $$$c=\cos{\left(2 \right)}$$$ e $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$:

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

Reescreva a tangente hiperbólica como $$$\tanh\left(\eta\right)=\frac{\sinh\left(\eta\right)}{\cosh\left(\eta\right)}$$$:

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

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

Então $$$du=\left(\cosh{\left(\eta \right)}\right)^{\prime }d\eta = \sinh{\left(\eta \right)} d\eta$$$ (veja os passos »), e obtemos $$$\sinh{\left(\eta \right)} d\eta = du$$$.

A integral torna-se

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

A integral de $$$\frac{1}{u}$$$ é $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Recorde que $$$u=\cosh{\left(\eta \right)}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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