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

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

Wende die Konstantenfaktorregel $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$ mit $$$c=\frac{\cos{\left(2 \right)}}{2}$$$ und $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$ an:

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

Schreibe den hyperbolischen Tangens als $$$\tanh\left(\eta\right)=\frac{\sinh\left(\eta\right)}{\cosh\left(\eta\right)}$$$ um.:

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

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

Dann $$$du=\left(\cosh{\left(\eta \right)}\right)^{\prime }d\eta = \sinh{\left(\eta \right)} d\eta$$$ (die Schritte sind » zu sehen), und es gilt $$$\sinh{\left(\eta \right)} d\eta = du$$$.

Also,

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

Das Integral von $$$\frac{1}{u}$$$ ist $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Zur Erinnerung: $$$u=\cosh{\left(\eta \right)}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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