integrated rate law
A speedometer tells you how fast you are going right now, but to know where you will be in an hour you need to add up all those speeds over time. An ordinary rate law is the speedometer — it gives the speed at one instant. The integrated rate law is the trip planner: it tells you how much reactant is left at any future moment.
Mathematically, the integrated rate law is what you get by integrating the differential rate law over time. It turns 'how fast' into 'how much remains when,' giving concentration as a direct function of elapsed time. Each reaction order has its own form: first order gives an exponential decay (and a straight line when you plot the logarithm of concentration against time), second order and zero order give different shapes, each with its own straight-line plot.
This matters in two big ways. Practically, it lets you forecast concentrations and compute half-lives without re-measuring at every instant. Diagnostically, you can test which order a reaction follows by seeing which of these plots comes out straight — the one that gives a line reveals the order, and its slope hands you the rate constant. The caveat is that these neat formulas assume a single, clean rate law; reactions with shifting mechanisms won't fit any one of them.
Suspect a reaction is first order? Plot ln[A] against time. If the points fall on a straight line, you have confirmed first order, the slope equals −k, and you can read off the concentration at any future time directly from the line — no need to run the experiment longer.
Which plot is straight tells you the order; the slope gives k.
Differential rate law (rate vs concentration) and integrated rate law (concentration vs time) describe the same reaction from two angles — don't mix up which one a problem is asking for.