Integral de $$$2 x^{\frac{3}{2}}$$$

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

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

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{3}{2}$$$:

$$2 {\color{red}{\int{x^{\frac{3}{2}} d x}}}=2 {\color{red}{\frac{x^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=2 {\color{red}{\left(\frac{2 x^{\frac{5}{2}}}{5}\right)}}$$

Portanto,

$$\int{2 x^{\frac{3}{2}} d x} = \frac{4 x^{\frac{5}{2}}}{5}$$

Adicione a constante de integração:

$$\int{2 x^{\frac{3}{2}} d x} = \frac{4 x^{\frac{5}{2}}}{5}+C$$

$$$\int 2 x^{\frac{3}{2}}\, dx = \frac{4 x^{\frac{5}{2}}}{5} + C$$$A