Integraal van $$$- 90 x^{2} - 9$$$

Bepaal $$$\int \left(- 90 x^{2} - 9\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(- 90 x^{2} - 9\right)d x}}} = {\color{red}{\left(- \int{9 d x} - \int{90 x^{2} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=9$$$:

$$- \int{90 x^{2} d x} - {\color{red}{\int{9 d x}}} = - \int{90 x^{2} d x} - {\color{red}{\left(9 x\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=90$$$ en $$$f{\left(x \right)} = x^{2}$$$:

$$- 9 x - {\color{red}{\int{90 x^{2} d x}}} = - 9 x - {\color{red}{\left(90 \int{x^{2} d x}\right)}}$$

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

$$- 9 x - 90 {\color{red}{\int{x^{2} d x}}}=- 9 x - 90 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 9 x - 90 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Dus,

$$\int{\left(- 90 x^{2} - 9\right)d x} = - 30 x^{3} - 9 x$$

Voeg de integratieconstante toe:

$$\int{\left(- 90 x^{2} - 9\right)d x} = - 30 x^{3} - 9 x+C$$

$$$\int \left(- 90 x^{2} - 9\right)\, dx = \left(- 30 x^{3} - 9 x\right) + C$$$A