Integralen av $$$- \frac{3}{x^{6}}$$$

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=-3$$$ och $$$f{\left(x \right)} = \frac{1}{x^{6}}$$$:

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

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-6$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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