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

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

A entrada é reescrita como: $$$\int{x \sqrt{x^{3}} d x}=\int{x^{\frac{5}{2}} d x}$$$.

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{5}{2}$$$:

$${\color{red}{\int{x^{\frac{5}{2}} d x}}}={\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}={\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$

Portanto,

$$\int{x^{\frac{5}{2}} d x} = \frac{2 x^{\frac{7}{2}}}{7}$$

Adicione a constante de integração:

$$\int{x^{\frac{5}{2}} d x} = \frac{2 x^{\frac{7}{2}}}{7}+C$$

$$$\int x \sqrt{x^{3}}\, dx = \frac{2 x^{\frac{7}{2}}}{7} + C$$$A