Integrating $\int^b_a [x]\,dx+\int^b_a [-x]\,dx$

Question

I came across a question today...

Integrate $\int^b_a [x]\,dx+\int^b_a [-x]\,dx$ where [.] denotes greatest integer function is equal to

Now this question is not helpful for me because in that question limits are integers but here, $a$ and $b$ can be any real number.

I know how greatest integers work but I don't know how to integrate them. I even tried to plot their graphs but they didn't helped me either.

Answer 1

Hint: For all real numbers which are not an integer, $$ [x] + [-x] = -1 $$