What is the difference between $\int v_i\,dt$ and $\int v_f\,dt$?

Question

I'm starting physics, and for a simple exercise, I wanted to derive each of the kinematic equations with calculus. I want to go from $v_f = v_i + at$ to $\Delta x = v_0t + 1/2at^2$, so I just integrated both sides of the first equation with respect to time, but I don't know what the difference between $\int v_i\,dt$ and $\int v_f\,dt$ is. Do they become two types of $\Delta x$?