Integral von $$$\frac{10}{x^{\frac{7}{5}}}$$$

Bestimme $$$\int \frac{10}{x^{\frac{7}{5}}}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=10$$$ und $$$f{\left(x \right)} = \frac{1}{x^{\frac{7}{5}}}$$$ an:

$${\color{red}{\int{\frac{10}{x^{\frac{7}{5}}} d x}}} = {\color{red}{\left(10 \int{\frac{1}{x^{\frac{7}{5}}} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=- \frac{7}{5}$$$ an:

$$10 {\color{red}{\int{\frac{1}{x^{\frac{7}{5}}} d x}}}=10 {\color{red}{\int{x^{- \frac{7}{5}} d x}}}=10 {\color{red}{\frac{x^{- \frac{7}{5} + 1}}{- \frac{7}{5} + 1}}}=10 {\color{red}{\left(- \frac{5 x^{- \frac{2}{5}}}{2}\right)}}=10 {\color{red}{\left(- \frac{5}{2 x^{\frac{2}{5}}}\right)}}$$

Daher,

$$\int{\frac{10}{x^{\frac{7}{5}}} d x} = - \frac{25}{x^{\frac{2}{5}}}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{10}{x^{\frac{7}{5}}} d x} = - \frac{25}{x^{\frac{2}{5}}}+C$$

$$$\int \frac{10}{x^{\frac{7}{5}}}\, dx = - \frac{25}{x^{\frac{2}{5}}} + C$$$A