Integraal van $$$x^{2} y^{2} z^{2}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int x^{2} y^{2} z^{2}\, dx$$$.

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

$${\color{red}{\int{x^{2} y^{2} z^{2} d x}}} = {\color{red}{y^{2} z^{2} \int{x^{2} d x}}}$$

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

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

Dus,

$$\int{x^{2} y^{2} z^{2} d x} = \frac{x^{3} y^{2} z^{2}}{3}$$

Voeg de integratieconstante toe:

$$\int{x^{2} y^{2} z^{2} d x} = \frac{x^{3} y^{2} z^{2}}{3}+C$$

$$$\int x^{2} y^{2} z^{2}\, dx = \frac{x^{3} y^{2} z^{2}}{3} + C$$$A