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
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.