Integral von $$$\frac{x^{2}}{2}$$$

Bestimme $$$\int \frac{x^{2}}{2}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(x \right)} = x^{2}$$$ an:

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

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=2$$$ an:

$$\frac{{\color{red}{\int{x^{2} d x}}}}{2}=\frac{{\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{2}=\frac{{\color{red}{\left(\frac{x^{3}}{3}\right)}}}{2}$$

Daher,

$$\int{\frac{x^{2}}{2} d x} = \frac{x^{3}}{6}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{x^{2}}{2} d x} = \frac{x^{3}}{6}+C$$

$$$\int \frac{x^{2}}{2}\, dx = \frac{x^{3}}{6} + C$$$A