Integraal van $$$4 x e^{3}$$$

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

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

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

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

$$4 e^{3} {\color{red}{\int{x d x}}}=4 e^{3} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=4 e^{3} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Dus,

$$\int{4 x e^{3} d x} = 2 x^{2} e^{3}$$

Voeg de integratieconstante toe:

$$\int{4 x e^{3} d x} = 2 x^{2} e^{3}+C$$

$$$\int 4 x e^{3}\, dx = 2 x^{2} e^{3} + C$$$A