Integraal van $$$- 2 x^{2}$$$

Bepaal $$$\int \left(- 2 x^{2}\right)\, dx$$$.

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

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

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

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

Dus,

$$\int{\left(- 2 x^{2}\right)d x} = - \frac{2 x^{3}}{3}$$

Voeg de integratieconstante toe:

$$\int{\left(- 2 x^{2}\right)d x} = - \frac{2 x^{3}}{3}+C$$

$$$\int \left(- 2 x^{2}\right)\, dx = - \frac{2 x^{3}}{3} + C$$$A