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

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

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

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

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

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

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 $$$\sinh{\left(\eta \right)} d\eta = du$$$.

Somit,

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

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

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

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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