Integraal van $$$\frac{24 x^{2}}{e^{8}}$$$

Bepaal $$$\int \frac{24 x^{2}}{e^{8}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{24}{e^{8}}$$$ en $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{\frac{24 x^{2}}{e^{8}} d x}}} = {\color{red}{\left(\frac{24 \int{x^{2} d x}}{e^{8}}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

$$\frac{24 {\color{red}{\int{x^{2} d x}}}}{e^{8}}=\frac{24 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{e^{8}}=\frac{24 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{e^{8}}$$

Dus,

$$\int{\frac{24 x^{2}}{e^{8}} d x} = \frac{8 x^{3}}{e^{8}}$$

Voeg de integratieconstante toe:

$$\int{\frac{24 x^{2}}{e^{8}} d x} = \frac{8 x^{3}}{e^{8}}+C$$

$$$\int \frac{24 x^{2}}{e^{8}}\, dx = \frac{8 x^{3}}{e^{8}} + C$$$A