Accurate position change for accelerating objects

[Droctagonopus]

If an object is subject to different forces at different times and these forces are totally unpredictable (the force at any instant after the current time cannot be predicted). How would we make the position change due to acceleration as accurate as possible?

I have chosen a method but I've encountered a problem.
Use x_{2}=\frac{1}{2}at^{2}+vt+x_{1} where a is the instantaneous acceleration, v is the instantaneous velocity, x_{1} is the position at the current time and t is a very small time interval. However, I need a way to find the same position value after unequal time intervals. Meaning that if I take a single 2 ms interval in one case and two 1 ms intervals in another, the final value doesn't have to be too accurate but it has to be the same for both cases. Is there an efficient way to do this?

 

[mfb]

You will need some model how your accelerations look like. Different step sizes are not an issue with your formula, if you know an average acceleration within (small) timesteps.