The Integrated Rate Equation

The problem with the differential rate law. The rate law $\text{Rate}=k[A]^x$, written as $-\dfrac{d[A]}{dt}=k[A]^x$, is a differential equation -- it tells you the INSTANTANEOUS rate at whatever concentration currently prevails, but it cannot directly be used to answer very practical questions such as: how long will it take for a given fraction of A to react? or, what concentration of A will remain after a specific elapsed time t? …