Integralen av $$$\frac{4}{x^{8}}$$$

Bestäm $$$\int \frac{4}{x^{8}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=4$$$ och $$$f{\left(x \right)} = \frac{1}{x^{8}}$$$:

$${\color{red}{\int{\frac{4}{x^{8}} d x}}} = {\color{red}{\left(4 \int{\frac{1}{x^{8}} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-8$$$:

$$4 {\color{red}{\int{\frac{1}{x^{8}} d x}}}=4 {\color{red}{\int{x^{-8} d x}}}=4 {\color{red}{\frac{x^{-8 + 1}}{-8 + 1}}}=4 {\color{red}{\left(- \frac{x^{-7}}{7}\right)}}=4 {\color{red}{\left(- \frac{1}{7 x^{7}}\right)}}$$

Alltså,

$$\int{\frac{4}{x^{8}} d x} = - \frac{4}{7 x^{7}}$$

Lägg till integrationskonstanten:

$$\int{\frac{4}{x^{8}} d x} = - \frac{4}{7 x^{7}}+C$$

$$$\int \frac{4}{x^{8}}\, dx = - \frac{4}{7 x^{7}} + C$$$A