Integrating a curious function

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Homework Statement



I'm having some trouble trying to integrate the following function

Homework Equations



[itex]\int([x/(logx)]dx)[/itex]

The Attempt at a Solution



I have tried integration by parts but I get stuck with harder integrals. What I'd like to know is that this function could be integrated or not. :) I've tried using Wolfram Alpha for this particular case but my math level is way below the explanations given there.
 

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  • #2
Ray Vickson
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Homework Statement



I'm having some trouble trying to integrate the following function

Homework Equations



[itex]\int([x/(logx)]dx)[/itex]

The Attempt at a Solution



I have tried integration by parts but I get stuck with harder integrals. What I'd like to know is that this function could be integrated or not. :) I've tried using Wolfram Alpha for this particular case but my math level is way below the explanations given there.
Do you mean that the integrand is [itex] f(x) = x/ \log(x)[/itex] or do you mean [itex] f(x) = [x/ \log(x)] [/itex], where [itex] [\cdots][/itex] is the "greatest-integer function"? If you mean the former, Maple expresses the result in terms of the non-elementary function Ei (the exponential integral): [tex] \mbox{Ei}(y) = P\int_{-\infty}^y \frac{e^t}{t} dt, [/tex]
with P denoting the principal value integral.

RGV
 
  • #3
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Yes, I meant the former, the integrand is x/logx.

Thank you for the information! I'm looking up a bit in other websites what this Ei function is in more detail.
 
  • #4
Matterwave
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Are you integrating that through the whole real line? In that case you really do have a principal value integral because you are moving through a pole in the integrand.
 
  • #5
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Integration by parts is the way I would go.

Try both functions for u. Ya got a 50% chance that your 1st choice is the correct one :)
 
  • #6
Ray Vickson
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Integration by parts is the way I would go.

Try both functions for u. Ya got a 50% chance that your 1st choice is the correct one :)
Integration by parts in NOT the way to go.

Your second comment makes no sense: the OP is 100% sure of what he/she means. Anyway, the second form f(x)= [x/log(x)] (where [] = greatest-integer function) will not have an analytically expressible integral---think about why not.

RGV
 

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