Integraal van $$$x y^{3}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int x y^{3}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=y^{3}$$$ en $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{x y^{3} d x}}} = {\color{red}{y^{3} \int{x d x}}}$$

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

$$y^{3} {\color{red}{\int{x d x}}}=y^{3} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=y^{3} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Dus,

$$\int{x y^{3} d x} = \frac{x^{2} y^{3}}{2}$$

Voeg de integratieconstante toe:

$$\int{x y^{3} d x} = \frac{x^{2} y^{3}}{2}+C$$

$$$\int x y^{3}\, dx = \frac{x^{2} y^{3}}{2} + C$$$A