Integralen av $$$\frac{1}{x^{\frac{8}{3}}}$$$

Bestäm $$$\int \frac{1}{x^{\frac{8}{3}}}\, dx$$$.

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=- \frac{8}{3}$$$:

$${\color{red}{\int{\frac{1}{x^{\frac{8}{3}}} d x}}}={\color{red}{\int{x^{- \frac{8}{3}} d x}}}={\color{red}{\frac{x^{- \frac{8}{3} + 1}}{- \frac{8}{3} + 1}}}={\color{red}{\left(- \frac{3 x^{- \frac{5}{3}}}{5}\right)}}={\color{red}{\left(- \frac{3}{5 x^{\frac{5}{3}}}\right)}}$$

Alltså,

$$\int{\frac{1}{x^{\frac{8}{3}}} d x} = - \frac{3}{5 x^{\frac{5}{3}}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{x^{\frac{8}{3}}} d x} = - \frac{3}{5 x^{\frac{5}{3}}}+C$$

$$$\int \frac{1}{x^{\frac{8}{3}}}\, dx = - \frac{3}{5 x^{\frac{5}{3}}} + C$$$A