Integraal van $$$f^{2} x^{n}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int f^{2} x^{n}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=f^{2}$$$ en $$$f{\left(x \right)} = x^{n}$$$:

$${\color{red}{\int{f^{2} x^{n} d x}}} = {\color{red}{f^{2} \int{x^{n} d x}}}$$

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

$$f^{2} {\color{red}{\int{x^{n} d x}}}=f^{2} {\color{red}{\frac{x^{n + 1}}{n + 1}}}=f^{2} {\color{red}{\frac{x^{n + 1}}{n + 1}}}$$

Dus,

$$\int{f^{2} x^{n} d x} = \frac{f^{2} x^{n + 1}}{n + 1}$$

Voeg de integratieconstante toe:

$$\int{f^{2} x^{n} d x} = \frac{f^{2} x^{n + 1}}{n + 1}+C$$

$$$\int f^{2} x^{n}\, dx = \frac{f^{2} x^{n + 1}}{n + 1} + C$$$A