Integral von $$$\frac{\tan{\left(x \right)}}{4}$$$

Bestimme $$$\int \frac{\tan{\left(x \right)}}{4}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{1}{4}$$$ und $$$f{\left(x \right)} = \tan{\left(x \right)}$$$ an:

$${\color{red}{\int{\frac{\tan{\left(x \right)}}{4} d x}}} = {\color{red}{\left(\frac{\int{\tan{\left(x \right)} d x}}{4}\right)}}$$

Schreibe die Tangente als $$$\tan\left(x\right)=\frac{\sin\left(x\right)}{\cos\left(x\right)}$$$ um:

$$\frac{{\color{red}{\int{\tan{\left(x \right)} d x}}}}{4} = \frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}}{4}$$

Sei $$$u=\cos{\left(x \right)}$$$.

Dann $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$\sin{\left(x \right)} dx = - du$$$.

Das Integral lässt sich umschreiben als

$$\frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}}{4} = \frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{4}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=-1$$$ und $$$f{\left(u \right)} = \frac{1}{u}$$$ an:

$$\frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{4} = \frac{{\color{red}{\left(- \int{\frac{1}{u} d u}\right)}}}{4}$$

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

$$- \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{4} = - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{4}$$

Zur Erinnerung: $$$u=\cos{\left(x \right)}$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{4} = - \frac{\ln{\left(\left|{{\color{red}{\cos{\left(x \right)}}}}\right| \right)}}{4}$$

Daher,

$$\int{\frac{\tan{\left(x \right)}}{4} d x} = - \frac{\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{4}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\tan{\left(x \right)}}{4} d x} = - \frac{\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{4}+C$$

$$$\int \frac{\tan{\left(x \right)}}{4}\, dx = - \frac{\ln\left(\left|{\cos{\left(x \right)}}\right|\right)}{4} + C$$$A