Integral de $$$x^{5} e^{7}$$$

Encontre $$$\int x^{5} e^{7}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=e^{7}$$$ e $$$f{\left(x \right)} = x^{5}$$$:

$${\color{red}{\int{x^{5} e^{7} d x}}} = {\color{red}{e^{7} \int{x^{5} d x}}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=5$$$:

$$e^{7} {\color{red}{\int{x^{5} d x}}}=e^{7} {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=e^{7} {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Portanto,

$$\int{x^{5} e^{7} d x} = \frac{x^{6} e^{7}}{6}$$

Adicione a constante de integração:

$$\int{x^{5} e^{7} d x} = \frac{x^{6} e^{7}}{6}+C$$

$$$\int x^{5} e^{7}\, dx = \frac{x^{6} e^{7}}{6} + C$$$A