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

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{\sqrt{2}}{2}$$$ och $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$:

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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