Integraal van $$$i^{\alpha} f g k^{\beta} x y$$$ met betrekking tot $$$x$$$

Bepaal $$$\int i^{\alpha} f g k^{\beta} x y\, dx$$$.

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

$${\color{red}{\int{i^{\alpha} f g k^{\beta} x y d x}}} = {\color{red}{i^{\alpha} f g k^{\beta} y \int{x d x}}}$$

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

$$i^{\alpha} f g k^{\beta} y {\color{red}{\int{x d x}}}=i^{\alpha} f g k^{\beta} y {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=i^{\alpha} f g k^{\beta} y {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Dus,

$$\int{i^{\alpha} f g k^{\beta} x y d x} = \frac{i^{\alpha} f g k^{\beta} x^{2} y}{2}$$

Voeg de integratieconstante toe:

$$\int{i^{\alpha} f g k^{\beta} x y d x} = \frac{i^{\alpha} f g k^{\beta} x^{2} y}{2}+C$$

$$$\int i^{\alpha} f g k^{\beta} x y\, dx = \frac{i^{\alpha} f g k^{\beta} x^{2} y}{2} + C$$$A