Integrating a curious function

[zip37]

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 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

 

[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.

 

[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.

 

[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 :)

 

[Ray Vickson]

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