Integraal van $$$8 x^{5}$$$

Bepaal $$$\int 8 x^{5}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=8$$$ en $$$f{\left(x \right)} = x^{5}$$$:

$${\color{red}{\int{8 x^{5} d x}}} = {\color{red}{\left(8 \int{x^{5} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=5$$$:

$$8 {\color{red}{\int{x^{5} d x}}}=8 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=8 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Dus,

$$\int{8 x^{5} d x} = \frac{4 x^{6}}{3}$$

Voeg de integratieconstante toe:

$$\int{8 x^{5} d x} = \frac{4 x^{6}}{3}+C$$

$$$\int 8 x^{5}\, dx = \frac{4 x^{6}}{3} + C$$$A