Integral von $$$\frac{2 \sin{\left(x \right)}}{5}$$$

Bestimme $$$\int \frac{2 \sin{\left(x \right)}}{5}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{2}{5}$$$ und $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ an:

$${\color{red}{\int{\frac{2 \sin{\left(x \right)}}{5} d x}}} = {\color{red}{\left(\frac{2 \int{\sin{\left(x \right)} d x}}{5}\right)}}$$

Das Integral des Sinus lautet $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$\frac{2 {\color{red}{\int{\sin{\left(x \right)} d x}}}}{5} = \frac{2 {\color{red}{\left(- \cos{\left(x \right)}\right)}}}{5}$$

Daher,

$$\int{\frac{2 \sin{\left(x \right)}}{5} d x} = - \frac{2 \cos{\left(x \right)}}{5}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{2 \sin{\left(x \right)}}{5} d x} = - \frac{2 \cos{\left(x \right)}}{5}+C$$

$$$\int \frac{2 \sin{\left(x \right)}}{5}\, dx = - \frac{2 \cos{\left(x \right)}}{5} + C$$$A