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Kurre

Member

Registered: 2006-07-18

Posts: 280

integer and decimal product integral

Let [x] be the greatest integer less than x, and let x' be the fraction part of x, ie x'=x-[x]. Let n be any integer ≥2. Evaluate the following integral in terms of n:

Last edited by Kurre (2008-06-25 06:42:29)

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: integer and decimal product integral

1≤x<2 [x]x'x = x(x-1)
2≤x<3 [x]x'x = 2x(x-2)
3≤x<4 [x]x'x = 3x(x-3)
...

A quick test shows that this does infact work!  (yaaay for me)

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Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: integer and decimal product integral

Good work My solution was slightly different (with x'=u for easier reading):

substituting x with x=u+k, we have dx/du=1 -> dx=du, and if x=k=u+k, then u=0, and if x=k+1=u+k then u=1, so we get

Last edited by Kurre (2008-06-27 01:40:45)