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Integrating X^x, Closed Form

Date: 10/31/96 at 15:6:46
From: Simon Judes
Subject: Integration of x^x and solution of y= xcosx

(1) How would you go about integrating x^x? I don't have a clue where 
to start.

Also, how do you express the equation y = xcosx in terms of y?

For both these problems, I have asked other people who have told me
that there is no simple solution. If this is the case, please send
the solution anyway, or at least give me some idea of the theory on
which it is based.


Date: 11/1/96 at 8:10:56
From: Doctor Jerry
Subject: Re: Integration of x^x and solution of y= xcosx

Dear Simon Judes,

(1) There is an algorithm due to a contemporary mathematician named 
Risch which 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 - which 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

Extensive tables of the error function would not exist 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. 

(2) As to solving the equation y = xcosx for x in terms of y, this 
can't be done in "closed form" or "by classical methods."  Both of 
these phrases are used to mean that this equation can't be solved by 
algebraic methods or with pencil and paper methods.  Of course, given 
a value of y, one can determine numerically (using Newton's method, 
for example) the values of x for which xcosx = y.

I'd like to give a parallel question to y = xcos x.  Can you solve the 
equation y = sinx for x in terms of y?  If you say yes and write 
x = arcsin y, then you have only recognized that the function inverse 
to sine is a well known function and you know its name.  You haven't 
actually found a finite expression for x.  The fact is, neither sine 
nor arcsin are known in the sense that y = x^2 or x = sqrt(y) is 
known.  Except for a few special cases, specific values of sine and 
arcsin must be looked up in a table or calculated numerically. 

-Doctor Jerry,  The Math Forum
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Associated Topics:
High School Calculus
High School Functions

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