Integralen av $$$\frac{1}{25 x^{6}}$$$

Bestäm $$$\int \frac{1}{25 x^{6}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{25}$$$ och $$$f{\left(x \right)} = \frac{1}{x^{6}}$$$:

$${\color{red}{\int{\frac{1}{25 x^{6}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x^{6}} d x}}{25}\right)}}$$

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

$$\frac{{\color{red}{\int{\frac{1}{x^{6}} d x}}}}{25}=\frac{{\color{red}{\int{x^{-6} d x}}}}{25}=\frac{{\color{red}{\frac{x^{-6 + 1}}{-6 + 1}}}}{25}=\frac{{\color{red}{\left(- \frac{x^{-5}}{5}\right)}}}{25}=\frac{{\color{red}{\left(- \frac{1}{5 x^{5}}\right)}}}{25}$$

Alltså,

$$\int{\frac{1}{25 x^{6}} d x} = - \frac{1}{125 x^{5}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{25 x^{6}} d x} = - \frac{1}{125 x^{5}}+C$$

$$$\int \frac{1}{25 x^{6}}\, dx = - \frac{1}{125 x^{5}} + C$$$A