Integraal van $$$x y$$$ met betrekking tot $$$y$$$

Bepaal $$$\int x y\, dy$$$.

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

$${\color{red}{\int{x y d y}}} = {\color{red}{x \int{y d y}}}$$

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

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

Dus,

$$\int{x y d y} = \frac{x y^{2}}{2}$$

Voeg de integratieconstante toe:

$$\int{x y d y} = \frac{x y^{2}}{2}+C$$

$$$\int x y\, dy = \frac{x y^{2}}{2} + C$$$A