Integrating a curious function

1. Nov 7, 2011

zip37

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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.

2. Nov 7, 2011

Ray Vickson

Do you mean that the integrand is $f(x) = x/ \log(x)$ or do you mean $f(x) = [x/ \log(x)]$, where $[\cdots]$ 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): $$\mbox{Ei}(y) = P\int_{-\infty}^y \frac{e^t}{t} dt,$$
with P denoting the principal value integral.

RGV

3. Nov 7, 2011

zip37

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. Nov 7, 2011

Matterwave

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. Nov 8, 2011

kmacinto

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. Nov 8, 2011

Ray Vickson

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