Integral von $$$c \cot{\left(x \right)}$$$ nach $$$x$$$

Bestimme $$$\int c \cot{\left(x \right)}\, dx$$$.

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

$${\color{red}{\int{c \cot{\left(x \right)} d x}}} = {\color{red}{c \int{\cot{\left(x \right)} d x}}}$$

Schreibe den Kotangens als $$$\cot\left(x\right)=\frac{\cos\left(x\right)}{\sin\left(x\right)}$$$ um:

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

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

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

Das Integral lässt sich umschreiben als

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

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

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

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

$$c \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = c \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)}$$

Daher,

$$\int{c \cot{\left(x \right)} d x} = c \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{c \cot{\left(x \right)} d x} = c \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$$

$$$\int c \cot{\left(x \right)}\, dx = c \ln\left(\left|{\sin{\left(x \right)}}\right|\right) + C$$$A