② recall that v = dx/dt, so we can do a nifty little rearrangement, to get:

Notice that as we change the variable we’re integrating over, we have to change the limits to match. In this case, we when our object was still and hadn’t moved (so v=0 and x=0).  After time t, our object has accelerated to velocity v and moved a distance x: hence the limits we’ve used.