# An ambiguous integral involving logarithm

• Jul 26th 2010, 09:42 AM
Heirot
An ambiguous integral involving logarithm
I get two different solutions for the integral

$\displaystyle \int_0^1 \frac{dx}{x} \ln \left( \frac{1+x}{1-x} \right)$ depending on how I evaluate it. Series expansion gives $\displaystyle \frac{\pi^2}{4}$ while Mathematica gives $\displaystyle 1+2 \ln 2$ as an analytical as well as a numerical solution. On the other hand, Wolfram Alpha also gives $\displaystyle \frac{\pi^2}{4}$. I'm thinking that there must be some problem with singularities and/or (non)uniform convergence of the Taylor series for the logarithm.

Any thoughts will be much appreciated.
• Jul 26th 2010, 09:51 AM
Ackbeet
Careful where your dx is, there. Can you give the command in Mathematica that gave you 1 + 2 ln(2)? I'm getting the first answer in Mathematica.
• Jul 26th 2010, 10:06 AM
Heirot
Oops, I mistyped the integral in Mathematica. Silly me. It is indeed $\displaystyle \frac{\pi^2}{4}$.

Thank you!
• Jul 26th 2010, 10:07 AM
Ackbeet
I figured.

You're very welcome. Have a good one!