Integraal van $$$9 \sqrt[8]{x}$$$

Bepaal $$$\int 9 \sqrt[8]{x}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=9$$$ en $$$f{\left(x \right)} = \sqrt[8]{x}$$$:

$${\color{red}{\int{9 \sqrt[8]{x} d x}}} = {\color{red}{\left(9 \int{\sqrt[8]{x} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=\frac{1}{8}$$$:

$$9 {\color{red}{\int{\sqrt[8]{x} d x}}}=9 {\color{red}{\int{x^{\frac{1}{8}} d x}}}=9 {\color{red}{\frac{x^{\frac{1}{8} + 1}}{\frac{1}{8} + 1}}}=9 {\color{red}{\left(\frac{8 x^{\frac{9}{8}}}{9}\right)}}$$

Dus,

$$\int{9 \sqrt[8]{x} d x} = 8 x^{\frac{9}{8}}$$

Voeg de integratieconstante toe:

$$\int{9 \sqrt[8]{x} d x} = 8 x^{\frac{9}{8}}+C$$

$$$\int 9 \sqrt[8]{x}\, dx = 8 x^{\frac{9}{8}} + C$$$A