Integral von $$$5 y^{2} \cos{\left(x \right)}$$$ nach $$$x$$$

Bestimme $$$\int 5 y^{2} \cos{\left(x \right)}\, dx$$$.

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

$${\color{red}{\int{5 y^{2} \cos{\left(x \right)} d x}}} = {\color{red}{\left(5 y^{2} \int{\cos{\left(x \right)} d x}\right)}}$$

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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