Integraal van $$$- x - 1$$$

Bepaal $$$\int \left(- x - 1\right)\, dx$$$.

Integreer termgewijs:

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

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

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

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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