The integral of ln x

To use parts to compute the integral of ln x, I have to divide the stuff in the integral up into two chunks. The only thing in the integral is ln x, so I use "ln x" and "1" as my chunks. I'll let u = ln x and dv = (1) dx; you can check that if you switch them, it won't work.

There is no reason to continue the table any further, because the powers of x just get larger. The first term comes from the usual parts pattern; I stop the table by integrating across the last row.

In this problem, I still have an integral to compute. Integration by parts often exchanges one integral for another; you're making progress if the new integral is easier to compute. In this case, it is:

This example illustrates another situation where parts is useful. The integrand ln x is a "single chunk"; I can't integrate it directly, and I don't see any obvious algebraic manipulations I could use to change the integral. Think of using parts when the integral contains a "single chunk" that can't be integrated directly or manipulated algebraically.

• Introduction: Integration by parts
• How do you set up an integration by parts table?
• The integral of x^2 e^x
  • How do you choose what to put in the table?
• The integral of ln x
• The integral of e^(2x) cos (3x)
• A recursion formula for the integral of (sin x)^n
• Computing a definite integral using integration by parts

Send comments about this page to: bikenaga@marauder.millersville.edu.

Last updated: June 13, 2005

Bruce Ikenaga's Home Page

Math Department Home Page

Millersville University Home Page

Copyright 1998 by Bruce Ikenaga