The Impossibility of Integrating x^x

Date: 08/07/98 at 09:47:58
From: Nick Black
Subject: Symbolic integration of x^x...impossible?

Hello. I am a 17 year-old at Walton High School, and last year I took
AP Calculus BC (covering differential, integral, and series-based
calculus). I found it an enjoyable and intriguing class, but there was 
one problem I could not solve. Misreading a question as asking for the 
symbolic integral of x^x, as opposed to the differential, I spent most 
of the day trying to generate a true integral. The differential was 
easy, of course. It equals x^x(ln(x)+1). 

In any case, I have tried converting the equation to a Taylor 
polynomial and creating a factorial-based polyterminal equation, 
converting to a summation, and generating a Rienmann sum, but ater a 
while, without seeing any discernible pattern, I get bogged down in the 
math (the 8th-order differential of x^x is an ugly beast). 

I hope this is within the realm of your service.  Thank you.

Date: 08/09/98 at 08:34:48
From: Doctor Jerry
Subject: Re: Symbolic integration of x^x...impossible?

Hi Nick,

There is an algorithm attributed to a contemporary mathematician named 
Risch that can decide whether the anti-derivative of a continuous 
function f can be expressed as a finite combination of elementary 
functions. The elementary functions include polynomials, the trig 
functions, the inverse trig functions, the exponential function and its 
inverse, etc.

There are many functions - called special functions - that fail to have 
an anti-derivative expressible as a finite combination of elementary 
functions. The so-called elliptic functions, the error function, and 
the gamma function are a few examples. The error function, which is 
extremely useful in both physics and statistics, is defined as:

   erf(x) = (2/sqrt(pi)) integral from 0 to x of e^(-t^2)dt

There would not exist extensive tables of the error function if the 
anti-derivative of e^(-t^2) were expressible as a finite combination 
of elementary functions. 

The anti-derivative of x^x is not expressible as a finite combination 
of elementary functions. I'm not sufficiently familiar with the Risch 
algorithm to even hint at a proof of it. 

- Doctor Jerry, The Math Forum
Check out our web site! http://mathforum.org/dr.math/