Integralen av $$$\frac{1}{x^{\frac{5}{2}}}$$$

Bestäm $$$\int \frac{1}{x^{\frac{5}{2}}}\, dx$$$.

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=- \frac{5}{2}$$$:

$${\color{red}{\int{\frac{1}{x^{\frac{5}{2}}} d x}}}={\color{red}{\int{x^{- \frac{5}{2}} d x}}}={\color{red}{\frac{x^{- \frac{5}{2} + 1}}{- \frac{5}{2} + 1}}}={\color{red}{\left(- \frac{2 x^{- \frac{3}{2}}}{3}\right)}}={\color{red}{\left(- \frac{2}{3 x^{\frac{3}{2}}}\right)}}$$

Alltså,

$$\int{\frac{1}{x^{\frac{5}{2}}} d x} = - \frac{2}{3 x^{\frac{3}{2}}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{x^{\frac{5}{2}}} d x} = - \frac{2}{3 x^{\frac{3}{2}}}+C$$

$$$\int \frac{1}{x^{\frac{5}{2}}}\, dx = - \frac{2}{3 x^{\frac{3}{2}}} + C$$$A