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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{3}{2}$$$:

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

Therefore,

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

Add the constant of integration:

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

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