Greatest integer function?

Question

We define a function [x] (Greatest Integer Function) as
f(x) = [x] =
{ n, n ≤ x < n + 1, n is an integer

I correctly define the integral ∫[x]dx as
∫[x]dx =
{ nx - (n² + n)/2, n ≤ x < n + 1, n is an integer

Now, how can you explain the constant quantity -(n² + n)/2? Where does it come from?

^_^
thanks!^_^
...easy...
^_^

Answer 1

I don't think it should be there. The area under the GIF is equal to the constant area, or nx, from n to n+1. There should be no extra.

Answer 2

It's too early in the morning to think of mathmatics. It makes my head hurt. Wait until I get my head screwed on.....maybe 9:00AM or so.