Funktion $$$\cos{\left(2 \right)} \tanh{\left(\eta \right)}$$$ integraali

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$ käyttäen $$$c=\cos{\left(2 \right)}$$$ ja $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$:

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

Esitä hyperbolinen tangentti muodossa $$$\tanh\left(\eta\right)=\frac{\sinh\left(\eta\right)}{\cosh\left(\eta\right)}$$$:

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

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

Tällöin $$$du=\left(\cosh{\left(\eta \right)}\right)^{\prime }d\eta = \sinh{\left(\eta \right)} d\eta$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$\sinh{\left(\eta \right)} d\eta = du$$$.

Integraali voidaan kirjoittaa muotoon

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

Funktion $$$\frac{1}{u}$$$ integraali on $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Muista, että $$$u=\cosh{\left(\eta \right)}$$$:

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

Näin ollen,

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

Lisää integrointivakio:

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

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