# Integrating a curious function

## Homework Statement

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

## Homework Equations

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

## 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.

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

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

## Homework Equations

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

## 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 $f(x) = x/ \log(x)$ 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

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.

Matterwave
Gold Member
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.

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 :)

Ray Vickson
Homework Helper
Dearly Missed
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