Integraal van $$$\frac{x}{2} - 1$$$

Bepaal $$$\int \left(\frac{x}{2} - 1\right)\, dx$$$.

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

$$\int{\frac{x}{2} d x} - {\color{red}{\int{1 d x}}} = \int{\frac{x}{2} d x} - {\color{red}{x}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(x \right)} = x$$$:

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

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

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

Dus,

$$\int{\left(\frac{x}{2} - 1\right)d x} = \frac{x^{2}}{4} - x$$

Vereenvoudig:

$$\int{\left(\frac{x}{2} - 1\right)d x} = \frac{x \left(x - 4\right)}{4}$$

Voeg de integratieconstante toe:

$$\int{\left(\frac{x}{2} - 1\right)d x} = \frac{x \left(x - 4\right)}{4}+C$$

$$$\int \left(\frac{x}{2} - 1\right)\, dx = \frac{x \left(x - 4\right)}{4} + C$$$A