Integral de $$$\frac{24 x^{2}}{e^{8}}$$$

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{24}{e^{8}}$$$ e $$$f{\left(x \right)} = x^{2}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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