Integral of Ln

Date: 4/13/96 at 12:38:14
From: Anonymous
Subject: calculus - solving an integral

Dr. Math,

I have an integral that I need to solve symbolically if possible.
It is the integral from 0 to 1 of ln(1-x) / x.

The correct answer appears to be -pi^2 / 6.  I got this answer from
numerical integration using Romberg's method.  However, I have
an interest in integrals of this form in more complex form that are
impractical to solve numerically.  So I need a symbolic notation for
the antiderivative.  I have tried substitutions, etc. but can't work
it out.  Can you help?

Date: 6/28/96 at 8:54:13
From: Doctor Jerry
Subject: Re: calculus - solving an integral

The function f(x)=-int_0^x (ln(t)/(t-1) dt, which is closely related 
to your integral (lower limit is 0 and upper limit is x), is known 
as the dilogarithm function.  It can not be expressed in terms of 
"elementary functions".  

However, the Handbook of Mathematical Functions by Milton 
Abramowitz and Irene Stegun, also lists the identity

f(x)+f(1-x)=-ln(x)*ln(1-x) + pi^2/6

If you take the limit of this expression as x->1-, you will obtain 
your result.

-Doctor Jerry,  The Math Forum