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

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

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

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

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

$$2 {\color{red}{\int{x^{e} d x}}}=2 {\color{red}{\frac{x^{1 + e}}{1 + e}}}=2 {\color{red}{\frac{x^{1 + e}}{1 + e}}}$$

Portanto,

$$\int{2 x^{e} d x} = \frac{2 x^{1 + e}}{1 + e}$$

Adicione a constante de integração:

$$\int{2 x^{e} d x} = \frac{2 x^{1 + e}}{1 + e}+C$$

$$$\int 2 x^{e}\, dx = \frac{2 x^{1 + e}}{1 + e} + C$$$A