Integral de $$$a x^{2}$$$ con respecto a $$$x$$$

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=a$$$ y $$$f{\left(x \right)} = x^{2}$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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