Integralen av $$$\sqrt{y^{5}}$$$

Bestäm $$$\int \sqrt{y^{5}}\, dy$$$.

Inmatningen skrivs om: $$$\int{\sqrt{y^{5}} d y}=\int{y^{\frac{5}{2}} d y}$$$.

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

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

Alltså,

$$\int{y^{\frac{5}{2}} d y} = \frac{2 y^{\frac{7}{2}}}{7}$$

Lägg till integrationskonstanten:

$$\int{y^{\frac{5}{2}} d y} = \frac{2 y^{\frac{7}{2}}}{7}+C$$

$$$\int \sqrt{y^{5}}\, dy = \frac{2 y^{\frac{7}{2}}}{7} + C$$$A