Integral von $$$\frac{4 x}{5}$$$

Bestimme $$$\int \frac{4 x}{5}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{4}{5}$$$ und $$$f{\left(x \right)} = x$$$ an:

$${\color{red}{\int{\frac{4 x}{5} d x}}} = {\color{red}{\left(\frac{4 \int{x d x}}{5}\right)}}$$

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

$$\frac{4 {\color{red}{\int{x d x}}}}{5}=\frac{4 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{5}=\frac{4 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{5}$$

Daher,

$$\int{\frac{4 x}{5} d x} = \frac{2 x^{2}}{5}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{4 x}{5} d x} = \frac{2 x^{2}}{5}+C$$

$$$\int \frac{4 x}{5}\, dx = \frac{2 x^{2}}{5} + C$$$A