Integraal van $$$\frac{x^{3}}{18}$$$

Bepaal $$$\int \frac{x^{3}}{18}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{18}$$$ en $$$f{\left(x \right)} = x^{3}$$$:

$${\color{red}{\int{\frac{x^{3}}{18} d x}}} = {\color{red}{\left(\frac{\int{x^{3} d x}}{18}\right)}}$$

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

$$\frac{{\color{red}{\int{x^{3} d x}}}}{18}=\frac{{\color{red}{\frac{x^{1 + 3}}{1 + 3}}}}{18}=\frac{{\color{red}{\left(\frac{x^{4}}{4}\right)}}}{18}$$

Dus,

$$\int{\frac{x^{3}}{18} d x} = \frac{x^{4}}{72}$$

Voeg de integratieconstante toe:

$$\int{\frac{x^{3}}{18} d x} = \frac{x^{4}}{72}+C$$

$$$\int \frac{x^{3}}{18}\, dx = \frac{x^{4}}{72} + C$$$A