Simple Kinetics Clarification
Not all reactions can or should be represented with a double arrow.  The general-audience example is that you "can't unbake a cake".  You are right that there are instances that we assume that k2 is negligible (relative to k1) and therefore a reaction is treated as if it is reversible.

Here's a sample plot of how the concentrations of A for the irreversible and reversible reactions compare for k1 = k2 = 1.



No surprises here; the reversible reaction results in an equilibrium concentration of A. We can compare the concentration of A in each of the reactions as a function of kr:



Here I've plotted Log[k2] on the x-axis and the ratio of A in the irreversible reaction to the reversible reaction.  The different lines are at various times (in seconds).  I've kept k1 = 1 and the initial concentration 1 as well.  The two equations predict the same concentration of A in the limit of very small k2, and by very small I suspect this means at least a factor of 10^4 or more (not shown on the graph).

While you are correct that the integrated rate laws get much more complicated for reversible reactions, the one you are considering is still solvable in closed form.