Integraal van $$$3 - 6 x^{2}$$$

Bepaal $$$\int \left(3 - 6 x^{2}\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(3 - 6 x^{2}\right)d x}}} = {\color{red}{\left(\int{3 d x} - \int{6 x^{2} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=3$$$:

$$- \int{6 x^{2} d x} + {\color{red}{\int{3 d x}}} = - \int{6 x^{2} d x} + {\color{red}{\left(3 x\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=6$$$ en $$$f{\left(x \right)} = x^{2}$$$:

$$3 x - {\color{red}{\int{6 x^{2} d x}}} = 3 x - {\color{red}{\left(6 \int{x^{2} d x}\right)}}$$

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

$$3 x - 6 {\color{red}{\int{x^{2} d x}}}=3 x - 6 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=3 x - 6 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Dus,

$$\int{\left(3 - 6 x^{2}\right)d x} = - 2 x^{3} + 3 x$$

Vereenvoudig:

$$\int{\left(3 - 6 x^{2}\right)d x} = x \left(3 - 2 x^{2}\right)$$

Voeg de integratieconstante toe:

$$\int{\left(3 - 6 x^{2}\right)d x} = x \left(3 - 2 x^{2}\right)+C$$

$$$\int \left(3 - 6 x^{2}\right)\, dx = x \left(3 - 2 x^{2}\right) + C$$$A