I have to prove that the improper integral ∫ ln(x)/(1-x) dx on the interval [0,1] is convergent.
I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1.
The Attempt at a Solution
The function can be approximated to ln(x) when it approaches zero. It is continuous therefore the integral is convergent on [0, 1/2].
We can also write that ln(x)=ln(1+x-1) ≅x-1, this may be applied when the finction is close to zero too.
However I have no idea how to prove that it is convergent on [1/2, 1].
I will greatly appreciate any help.