Why Can't Some Functions be Integrated?

Date: 11/06/96 at 15:49:26
From: Deepak Shenoy
Subject: Calculus - Integration

Why can't some functions be integrated? For example, I have been 
trying to evaluate Integral[x tanx dx] (without limits) and have not 
been able to do it. I tried the regular uv rule where you integrate 
the trigonometric function first. 

I have been told that this integral can be found only by integrating 
within limits or by expanding the function into a polynomial. Please 
advise.

Date: 11/07/96 at 08:22:27
From: Doctor Jerry
Subject: Re: Calculus - Integration

Why can't some functions be integrated?  For the same reason that an 
arbitrary angle can't be trisected or there is no general formula for 
finding the roots of a fifth degree polynomial.  There is a proof that 
not all functions can be integrated, just as there is a proof that not 
all angles can be trisected (with straightedge and compass), and a 
proof that there is no formula for finding the roots of a fifth degree 
polynomial. Of these proofs, the first and third are difficult, 
requiring courses beyond those commonly given in the first four years 
of undergraduate mathematics (in the U.S.A.).

There is a procedure due to a contemporary mathematician named Risch 
which allows you to determine 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.

The computer algebra system Mathematica expresses the anti-derivative 
of x*tan(x) in terms of the polylogarithm function, which is not an 
elementary function.  This is not a proof that an anti-derivative of 
x*tan(x) can't be expressed as a finite combination of elementary 
functions, but a clue that this might well be the case. 

Integration by parts was a reasonable thing to try, but as you found, 
it doesn't seem to work out.  The other standard techniques also fail.

You say that x*tan(x) can be integrated within limits.  If you mean by 
this that the integral can be done numerically, by using Simpson's 
Rule, for example, I agree.  Whether there are limits or not doesn't 
change the fact that no elementary anti-derivative is known for this 
function.  Numerical integration is thus needed.  

The other alternative you mentioned is to expand x*tan(x) into a power 
series and integrate term-by-term.  This gives an infinite series for 
the anti-derivative.  This is a useful but roundabout way of solving 
the problem, not an expression of the anti-derivative in terms of a 
finite combination of elementary functions.

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

Date: 11/07/96 at 18:26:37
From: Doctor Anthony
Subject: Re: Calculus - Integration

This is a messy integral.  For |x|< pi/2, here is your power series 
approximation:  

INT[x.tan(x).dx] = x^3/3 + x^5/15 + (2/105)x^7 + (17/2835)x^9 + ...

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