Integral of $$$- x^{2}$$$

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

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

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

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

Therefore,

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

Add the constant of integration:

$$\int{\left(- x^{2}\right)d x} = - \frac{x^{3}}{3}+C$$

Answer: $$$\int{\left(- x^{2}\right)d x}=- \frac{x^{3}}{3}+C$$$