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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

Therefore,

$$\int{\frac{x^{2}}{2} d x} = \frac{x^{3}}{6}$$

Add the constant of integration:

$$\int{\frac{x^{2}}{2} d x} = \frac{x^{3}}{6}+C$$

$$$\int \frac{x^{2}}{2}\, dx = \frac{x^{3}}{6} + C$$$A