Integraal van $$$- a x$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(- a x\right)\, dx$$$.

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

$${\color{red}{\int{\left(- a x\right)d x}}} = {\color{red}{\left(- a \int{x d x}\right)}}$$

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

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

Dus,

$$\int{\left(- a x\right)d x} = - \frac{a x^{2}}{2}$$

Voeg de integratieconstante toe:

$$\int{\left(- a x\right)d x} = - \frac{a x^{2}}{2}+C$$

$$$\int \left(- a x\right)\, dx = - \frac{a x^{2}}{2} + C$$$A