Integraal van $$$x^{4} e^{4}$$$

Bepaal $$$\int x^{4} e^{4}\, dx$$$.

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

$${\color{red}{\int{x^{4} e^{4} d x}}} = {\color{red}{e^{4} \int{x^{4} d x}}}$$

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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