Integral de $$$x^{3} e^{2}$$$

Encontre $$$\int x^{3} e^{2}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=e^{2}$$$ e $$$f{\left(x \right)} = x^{3}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=3$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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