Integraal van $$$\frac{5 x^{4}}{3}$$$

Bepaal $$$\int \frac{5 x^{4}}{3}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{5}{3}$$$ en $$$f{\left(x \right)} = x^{4}$$$:

$${\color{red}{\int{\frac{5 x^{4}}{3} d x}}} = {\color{red}{\left(\frac{5 \int{x^{4} d x}}{3}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=4$$$:

$$\frac{5 {\color{red}{\int{x^{4} d x}}}}{3}=\frac{5 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}}{3}=\frac{5 {\color{red}{\left(\frac{x^{5}}{5}\right)}}}{3}$$

Dus,

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

Voeg de integratieconstante toe:

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

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