Integral de $$$\frac{31 \sqrt{x^{3}}}{x}$$$

Encontre $$$\int \frac{31 \sqrt{x^{3}}}{x}\, dx$$$.

A entrada é reescrita como: $$$\int{\frac{31 \sqrt{x^{3}}}{x} d x}=\int{31 \sqrt{x} d x}$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=31$$$ e $$$f{\left(x \right)} = \sqrt{x}$$$:

$${\color{red}{\int{31 \sqrt{x} d x}}} = {\color{red}{\left(31 \int{\sqrt{x} d x}\right)}}$$

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

$$31 {\color{red}{\int{\sqrt{x} d x}}}=31 {\color{red}{\int{x^{\frac{1}{2}} d x}}}=31 {\color{red}{\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}=31 {\color{red}{\left(\frac{2 x^{\frac{3}{2}}}{3}\right)}}$$

Portanto,

$$\int{31 \sqrt{x} d x} = \frac{62 x^{\frac{3}{2}}}{3}$$

Adicione a constante de integração:

$$\int{31 \sqrt{x} d x} = \frac{62 x^{\frac{3}{2}}}{3}+C$$

$$$\int \frac{31 \sqrt{x^{3}}}{x}\, dx = \frac{62 x^{\frac{3}{2}}}{3} + C$$$A