Find the following integral: $\int(ax^{2} + bx + c) dx$

To find the integral $\int(ax^{2} + bx + c) dx$,we use the linearity property of integration:
$\int(ax^{2} + bx + c) dx = a \int x^{2} dx + b \int x dx + c \int 1 dx$
Using the power rule $\int x^{n} dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$:
$= a \left(\frac{x^{3}}{3}\right) + b \left(\frac{x^{2}}{2}\right) + c(x) + C$
$= \frac{ax^{3}}{3} + \frac{bx^{2}}{2} + cx + C$
where $C$ is the constant of integration.