Integraal van $$$a^{2} x$$$ met betrekking tot $$$x$$$

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

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

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

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

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

Dus,

$$\int{a^{2} x d x} = \frac{a^{2} x^{2}}{2}$$

Voeg de integratieconstante toe:

$$\int{a^{2} x d x} = \frac{a^{2} x^{2}}{2}+C$$

$$$\int a^{2} x\, dx = \frac{a^{2} x^{2}}{2} + C$$$A