Finding the Indefinite Integral

Date: 2/1/96 at 17:10:41
From: Anonymous
Subject: integral calculus

Dr. Math,

	I am a senior in an advanced high school calculus class.  
I am trying to find the indefinite integral of the following 
function:

   f(x)=x^x

I have no clue as to what to do with this problem and I was 
wondering if you could help.

   -Matt Ford

Date: 5/29/96 at 1:29:23
From: Doctor Pete
Subject: Re: integral calculus

There isn't a "nice" solution to the integral of f(x)=x^x.  None 
of the substitutions or methods taught to evaluate integrals can 
completely integrate this function, and it is a general rule that 
most functions  don't have integrals that are in "closed form," 
that is, the integral is  some expression in other elementary 
functions.  A well-known function  with no closed form integral is 
e^(-x^2); however, the definite  (improper) integral from 0 to 
infinity is closed.  (Can you find its  value?)

On the other hand, one can compute the derivative f'(x) by 
logarithmic differentiation.  We have f(x) = x^x, so ln(f(x)) = 
x*ln(x) and d(ln(f(x)))/dx = f'(x)/f(x) = 1+ln(x), so f'(x) = 
(1+ln(x))*x^x.

-Doctor Pete,  The Math Forum