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.


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

  • $\begingroup$ ok understood but can we really integrate a greatest integer function from limits a to b $\endgroup$ – manshu Mar 5 '16 at 19:02
  • 2
    $\begingroup$ @manshu: It is Riemann/Lebesgue integrable (continuous with the exception of finitely many points), and therefore you can combine the two integrals into a single one for the sum. $\endgroup$ – Martin R Mar 5 '16 at 19:04
  • $\begingroup$ So it means that we can't solve $\int^b_a [x]\,dx$ ? $\endgroup$ – manshu Mar 5 '16 at 19:10
  • $\begingroup$ @manshu: If you mean "solve by writing an explicit formula": Yes, you can do that. But you don't need it for this question. $\endgroup$ – Martin R Mar 5 '16 at 19:12

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