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

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

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

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

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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