Integral of ln(x)/(x+1) from 1 ot 0

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Discussion Overview

The discussion revolves around the evaluation of the integral of ln(x)/(x+1) from 0 to 1, exploring methods of integration and the existence of a primitive function. Participants address challenges related to improper integrals and the use of series expansions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in solving the integral due to an undefined term from ln(0) and seeks advice.
  • Another participant states that the integral does not have a primitive in terms of elementary functions, later clarifying that it is a definite integral.
  • A different participant suggests that the issue with ln(0) indicates the need to consider improper integrals.
  • One participant proposes an integration by parts approach, leading to a transformation of the integral but concludes that the resulting integral is not integrable in elementary terms, suggesting a value of (π²)/6 for a related integral.
  • Another participant agrees that the integral can be evaluated but reiterates that no primitive exists in terms of elementary functions, suggesting the use of the power series for ln(x+1).
  • A participant questions the clarity of the discussion, pointing out a contradiction regarding which integral is being referenced.
  • A later reply defends the previous post, stating that the integrals discussed yield each other and mentioning the use of the Taylor Series for ln(1+x) to derive a series representation of the integral.

Areas of Agreement / Disagreement

Participants express differing views on the integrability of the integral in question and whether it can be evaluated using elementary functions. The discussion remains unresolved regarding the clarity of which integral is being discussed and the methods of evaluation.

Contextual Notes

There are unresolved assumptions regarding the treatment of improper integrals and the definitions of the integrals being discussed. The relationship between the integrals mentioned is also not fully clarified.

coki2000
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Hello,
what is it's solvation.I tried to solve it by parts but i found undefinable term(from ln(0)).Please give me an advice.Thanks.
 
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This integral doesn't have a primitive in terms of elementary functions.

Edit: didn't notice it was a definite integral.
 
coki2000 said:
Hello,
what is it's solvation.I tried to solve it by parts but i found undefinable term(from ln(0)).Please give me an advice.Thanks.
That should be a clue that you've forgotten about improper integrals...
 
If i integrete this integral by parts

[tex]\int_{0}^{1}\frac{ln(x+1)}{x}dx[/tex]

ln(x+1)=u ,1/(x+1)dx=du and 1/xdx=dv, lnx=v

[tex]\int_{0}^{1}\frac{ln(x+1)}{x}dx=ln(x+1)lnx\mid_{0}^{1}-\int_{0}^{1} \frac{lnx}{x+1}dx[/tex]

This time integral of ln(x+1)/x from 0 to 1 is not integratable but answer of this is (pi^2)/6.
 
You can integrate it just fine, however there does not exist a primitive in terms of elementary functions. You can evaluate this integral by using the power series of ln(x+1).
 
The message #4 is in contradiction with the title of the thread…
Which integral is it ?
 
guerom00 said:
The message #4 is in contradiction with the title of the thread…
Which integral is it ?

No it isn't. The post showed that each integral yields the other. And as Cyosis mentioned, using the Taylor Series for ln(1+x) centered at x=0, we can get a series representation of the integral, and it so happens in this case it simplifies to a simple combination of well known constants.
 

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