Integraal van $$$2 x - 12$$$

Bepaal $$$\int \left(2 x - 12\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(2 x - 12\right)d x}}} = {\color{red}{\left(- \int{12 d x} + \int{2 x d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=12$$$:

$$\int{2 x d x} - {\color{red}{\int{12 d x}}} = \int{2 x d x} - {\color{red}{\left(12 x\right)}}$$

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

$$- 12 x + {\color{red}{\int{2 x d x}}} = - 12 x + {\color{red}{\left(2 \int{x d x}\right)}}$$

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

$$- 12 x + 2 {\color{red}{\int{x d x}}}=- 12 x + 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 12 x + 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Dus,

$$\int{\left(2 x - 12\right)d x} = x^{2} - 12 x$$

Vereenvoudig:

$$\int{\left(2 x - 12\right)d x} = x \left(x - 12\right)$$

Voeg de integratieconstante toe:

$$\int{\left(2 x - 12\right)d x} = x \left(x - 12\right)+C$$

$$$\int \left(2 x - 12\right)\, dx = x \left(x - 12\right) + C$$$A