Integralen av $$$\frac{1}{6 x^{7}}$$$

Bestäm $$$\int \frac{1}{6 x^{7}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{6}$$$ och $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$:

$${\color{red}{\int{\frac{1}{6 x^{7}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x^{7}} d x}}{6}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-7$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{7}} d x}}}}{6}=\frac{{\color{red}{\int{x^{-7} d x}}}}{6}=\frac{{\color{red}{\frac{x^{-7 + 1}}{-7 + 1}}}}{6}=\frac{{\color{red}{\left(- \frac{x^{-6}}{6}\right)}}}{6}=\frac{{\color{red}{\left(- \frac{1}{6 x^{6}}\right)}}}{6}$$

Alltså,

$$\int{\frac{1}{6 x^{7}} d x} = - \frac{1}{36 x^{6}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{6 x^{7}} d x} = - \frac{1}{36 x^{6}}+C$$

$$$\int \frac{1}{6 x^{7}}\, dx = - \frac{1}{36 x^{6}} + C$$$A