Integral von $$$2 x^{\frac{3}{2}}$$$

Bestimme $$$\int 2 x^{\frac{3}{2}}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=2$$$ und $$$f{\left(x \right)} = x^{\frac{3}{2}}$$$ an:

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

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=\frac{3}{2}$$$ an:

$$2 {\color{red}{\int{x^{\frac{3}{2}} d x}}}=2 {\color{red}{\frac{x^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=2 {\color{red}{\left(\frac{2 x^{\frac{5}{2}}}{5}\right)}}$$

Daher,

$$\int{2 x^{\frac{3}{2}} d x} = \frac{4 x^{\frac{5}{2}}}{5}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{2 x^{\frac{3}{2}} d x} = \frac{4 x^{\frac{5}{2}}}{5}+C$$

$$$\int 2 x^{\frac{3}{2}}\, dx = \frac{4 x^{\frac{5}{2}}}{5} + C$$$A