Integralen av $$$\frac{1}{x^{\frac{7}{5}}}$$$

Bestäm $$$\int \frac{1}{x^{\frac{7}{5}}}\, dx$$$.

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=- \frac{7}{5}$$$:

$${\color{red}{\int{\frac{1}{x^{\frac{7}{5}}} d x}}}={\color{red}{\int{x^{- \frac{7}{5}} d x}}}={\color{red}{\frac{x^{- \frac{7}{5} + 1}}{- \frac{7}{5} + 1}}}={\color{red}{\left(- \frac{5 x^{- \frac{2}{5}}}{2}\right)}}={\color{red}{\left(- \frac{5}{2 x^{\frac{2}{5}}}\right)}}$$

Alltså,

$$\int{\frac{1}{x^{\frac{7}{5}}} d x} = - \frac{5}{2 x^{\frac{2}{5}}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{x^{\frac{7}{5}}} d x} = - \frac{5}{2 x^{\frac{2}{5}}}+C$$

$$$\int \frac{1}{x^{\frac{7}{5}}}\, dx = - \frac{5}{2 x^{\frac{2}{5}}} + C$$$A