Integraal van $$$1 - y$$$

Bepaal $$$\int \left(1 - y\right)\, dy$$$.

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dy = c y$$$ toe met $$$c=1$$$:

$$- \int{y d y} + {\color{red}{\int{1 d y}}} = - \int{y d y} + {\color{red}{y}}$$

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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