# Evaluating the integral $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx$

I don't know how to deal with this integral:

$$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx,$$ where z is a complex number.

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@TMM: I think some tags related to complex numbers are needed. –  B. S. Apr 2 '13 at 18:23
@BabakS. Certainly you are able to add those tags as well :) –  TMM Apr 2 '13 at 18:26

Let $f(z)=\int_{-1}^1\dfrac{1}{\sqrt{1-x^2}}\ln|z-x|~dx$ ,
Then $\dfrac{df(z)}{dz}=\int_{-1}^1\dfrac{1}{(z-x)\sqrt{1-x^2}}dx$
$\dfrac{df(z)}{dz}=\dfrac{\pi}{\sqrt{z^2-1}}$
$f(z)=\int\dfrac{\pi}{\sqrt{z^2-1}}dz=\ln(z+\sqrt{z^2-1})+C$
But I don't know how to find $C$.