Velocity is the rate of change of position, defined as $v = \frac{dx}{dt}$ . To find the distance travelled (displacement) between two time points, we integrate the velocity function with respect to time.

1. Set up the integral: $x = \int_{t_1}^{t_2} v \, dt = \int_{1}^{2} (At + Bt^2) \, dt$

2. Integrate: Using the power rule for integration $\int t^n dt = \frac{t^{n+1}}{n+1}$: $x = \left[ \frac{At^2}{2} + \frac{Bt^3}{3} \right]_{1}^{2}$

3. Apply limits: Upper limit ($t=2$): $\frac{A(2)^2}{2} + \frac{B(2)^3}{3} = \frac{4A}{2} + \frac{8B}{3} = 2A + \frac{8B}{3}$ Lower limit ($t=1$): $\frac{A(1)^2}{2} + \frac{B(1)^3}{3} = \frac{A}{2} + \frac{B}{3}$

4. Calculate the difference: $x = \left( 2A + \frac{8B}{3} \right) - \left( \frac{A}{2} + \frac{B}{3} \right)$ $x = \left( 2A - \frac{A}{2} \right) + \left( \frac{8B}{3} - \frac{B}{3} \right)$ $x = \frac{3A}{2} + \frac{7B}{3}$