Integraal van $$$\frac{3^{\frac{2}{3}}}{3 \sqrt[3]{x}}$$$

Bepaal $$$\int \frac{3^{\frac{2}{3}}}{3 \sqrt[3]{x}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{3^{\frac{2}{3}}}{3}$$$ en $$$f{\left(x \right)} = \frac{1}{\sqrt[3]{x}}$$$:

$${\color{red}{\int{\frac{3^{\frac{2}{3}}}{3 \sqrt[3]{x}} d x}}} = {\color{red}{\left(\frac{3^{\frac{2}{3}} \int{\frac{1}{\sqrt[3]{x}} d x}}{3}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=- \frac{1}{3}$$$:

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

Dus,

$$\int{\frac{3^{\frac{2}{3}}}{3 \sqrt[3]{x}} d x} = \frac{3^{\frac{2}{3}} x^{\frac{2}{3}}}{2}$$

Voeg de integratieconstante toe:

$$\int{\frac{3^{\frac{2}{3}}}{3 \sqrt[3]{x}} d x} = \frac{3^{\frac{2}{3}} x^{\frac{2}{3}}}{2}+C$$

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