Integral of Ln x

Date: 3/31/96 at 13:28:34
From: Anonymous
Subject: Calculus

Dr. Math, 

Can you please tell me what the integral of the natural log of 
x is (ln x)?

Date: 3/31/96 at 22:27:25
From: Doctor Sebastien
Subject: Re: Calculus

Hello,

Here's the question: Integral (ln x) dx

Let's substitute ln x by y:

      x = e^y

  dx/dy = e^y

So 

  Integral (ln x) dx = Integral (ln x) dx/dy dy = Integral (y e^y) dy

Using integration by parts, 

  Integral (y e^y) dy = uv - Integral (v du/dy) dy

Let u = y.

        du/dy = 1

  Integral dv = Integral (e^y) dy

            v = e^y

Therefore, 

  Integral (y e^y) dy = y e^y - Integral (e^y * 1) dy
                 
                      = y e^y - (e^y) + c, 

where c is a constant. Therefore, 

  Integral (ln x) dx = y e^y - (e^y) + c

                     = (ln x)e^(ln x) - e^(ln x) + c

                     = x(ln x) - x + c

Sometimes you will do integrations of simple terms that don't 
have a solution that take only one line.  You will then have to use 
integration by parts.

-Doctor Sebastien,  The Math Forum