Integrating a curious function

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Homework Help Overview

The discussion revolves around the integration of the function \( \frac{x}{\log x} \). Participants are exploring whether this function can be integrated and discussing the implications of different interpretations of the integrand.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the use of integration by parts and express uncertainty about the integrability of the function. There is a clarification regarding the interpretation of the function, whether it is \( \frac{x}{\log x} \) or the greatest-integer function form. Questions about the nature of the integral, particularly regarding principal value integrals, are also raised.

Discussion Status

The discussion is active, with participants providing insights into the nature of the function and its integral. Some guidance on the use of integration by parts is mentioned, although there are conflicting opinions about its effectiveness. The conversation reflects a mix of interpretations and attempts to clarify the original poster's intent.

Contextual Notes

There is mention of potential confusion regarding the form of the integrand and the implications of integrating through a pole. The original poster expresses a lack of confidence in their mathematical understanding, which may affect the discussion's direction.

zip37
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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.
 
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zip37 said:

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.
 
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 :)
 
kmacinto said:
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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