If ∫_o^[u]x dx= ∫_o^u[x] dx, u>0, and [x] denotes the greatest integer < = x, is it necessarily true that u=[u]+ 1/2?

 LEFT INTEGRAL = [u]^2/2

 RIGHT INTEGRAL = [u](2u-([u]+1))/2

 LEFT INTEGRAL = RIGHT INTEGRAL 

              iff

      [u](2(u-[u])-1) = 0

Therefore, u = [u]+1/2 unless u<1 and u != 1/2.

Posted by Bractals on 2007-03-21 09:58:20