Well, it depends on whether or not acceleration is constant. If it is it is simply
s = ut+1/2 at^2
If a is not constant, you need to numerically integrated. Now there is a variety of methods and none of them will beat doing this by hand for accuracy, as they are all ultimately approximate solutions.
The easiest and least accurate is Euler's method . Here you divide time into discrete chunks called time steps, and perform
v[n] = v[n-1] * t * a[t]
n is index, t is size of a time step. Position is similarly updated. This is only really good for those cases where accuracy is not all that important. A special version of Euler's method will yield an exact solution for projectile motion (see wiki), so while this method is crude, it can be perfect for some suituations.
The most common numerical integration method used in games and in some chemistry simulations is Velocity Verlet, which is a special form of the more generic Verlet method. I would recommend this one if Euler's is too crude.
