Integral von $$$- \frac{3}{x^{6}}$$$

Bestimme $$$\int \left(- \frac{3}{x^{6}}\right)\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=-3$$$ und $$$f{\left(x \right)} = \frac{1}{x^{6}}$$$ an:

$${\color{red}{\int{\left(- \frac{3}{x^{6}}\right)d x}}} = {\color{red}{\left(- 3 \int{\frac{1}{x^{6}} d x}\right)}}$$

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=-6$$$ an:

$$- 3 {\color{red}{\int{\frac{1}{x^{6}} d x}}}=- 3 {\color{red}{\int{x^{-6} d x}}}=- 3 {\color{red}{\frac{x^{-6 + 1}}{-6 + 1}}}=- 3 {\color{red}{\left(- \frac{x^{-5}}{5}\right)}}=- 3 {\color{red}{\left(- \frac{1}{5 x^{5}}\right)}}$$

Daher,

$$\int{\left(- \frac{3}{x^{6}}\right)d x} = \frac{3}{5 x^{5}}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- \frac{3}{x^{6}}\right)d x} = \frac{3}{5 x^{5}}+C$$

$$$\int \left(- \frac{3}{x^{6}}\right)\, dx = \frac{3}{5 x^{5}} + C$$$A