Integraal van $$$150 - 60 x$$$

Bepaal $$$\int \left(150 - 60 x\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(150 - 60 x\right)d x}}} = {\color{red}{\left(\int{150 d x} - \int{60 x d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=150$$$:

$$- \int{60 x d x} + {\color{red}{\int{150 d x}}} = - \int{60 x d x} + {\color{red}{\left(150 x\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=60$$$ en $$$f{\left(x \right)} = x$$$:

$$150 x - {\color{red}{\int{60 x d x}}} = 150 x - {\color{red}{\left(60 \int{x d x}\right)}}$$

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

$$150 x - 60 {\color{red}{\int{x d x}}}=150 x - 60 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=150 x - 60 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Dus,

$$\int{\left(150 - 60 x\right)d x} = - 30 x^{2} + 150 x$$

Vereenvoudig:

$$\int{\left(150 - 60 x\right)d x} = 30 x \left(5 - x\right)$$

Voeg de integratieconstante toe:

$$\int{\left(150 - 60 x\right)d x} = 30 x \left(5 - x\right)+C$$

$$$\int \left(150 - 60 x\right)\, dx = 30 x \left(5 - x\right) + C$$$A