Integral of $$$a x^{2}$$$ with respect to $$$x$$$

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

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

$${\color{red}{\int{a x^{2} d x}}} = {\color{red}{a \int{x^{2} d x}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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