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Subtract Half, Get GI? (Posted on 2007-03-20) Difficulty: 2 of 5
If ∫o[u]x dx= ∫ou[x] dx, u>0, and [x] denotes the greatest integer < = x, is it necessarily true that u=[u]+ 1/2?

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

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Solution Solution Comment 2 of 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
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