Integraal van $$$9 \pi x^{2}$$$

Bepaal $$$\int 9 \pi x^{2}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=9 \pi$$$ en $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{9 \pi x^{2} d x}}} = {\color{red}{\left(9 \pi \int{x^{2} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

$$9 \pi {\color{red}{\int{x^{2} d x}}}=9 \pi {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=9 \pi {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Dus,

$$\int{9 \pi x^{2} d x} = 3 \pi x^{3}$$

Voeg de integratieconstante toe:

$$\int{9 \pi x^{2} d x} = 3 \pi x^{3}+C$$

$$$\int 9 \pi x^{2}\, dx = 3 \pi x^{3} + C$$$A