Integral von $$$\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}$$$

Bestimme $$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Schreiben Sie den Integranden um:

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

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

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

Das Integral des Kosinus ist $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$2 {\color{red}{\int{\cos{\left(x \right)} d x}}} = 2 {\color{red}{\sin{\left(x \right)}}}$$

Daher,

$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}+C$$

$$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx = 2 \sin{\left(x \right)} + C$$$A