Integralen av $$$x^{\frac{2}{3}}$$$

Bestäm $$$\int x^{\frac{2}{3}}\, dx$$$.

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{2}{3}$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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