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Re: A tough Integral

• To: mathgroup at smc.vnet.net
• Subject: [mg28555] Re: A tough Integral
• From: Richard Easther <easther at physics.columbia.edu>
• Date: Fri, 27 Apr 2001 03:56:09 -0400 (EDT)
• Organization: Columbia University
• References: <200104250530.BAA19306@smc.vnet.net> <9c7n4d$sda@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 25 Apr 2001, Daniel Lichtblau wrote:

>bobbym1953 wrote:
>> 
>> Does anyone know how I can get the following Integral to at least 60 places,
>> using Mathematica?
>> 
>> Int(1/(cos(x)+x^2)) between x=0 and x=infinity. Both Integrate and NIntegrate
>> seemed helpless. 
> 
> 60 places?! I wish you luck.

Actually, it can be done.  Break the integral into two pieces, I_1 where
the integral is integrated from O<x<N, and I_2 where we integrate between
N<=x<Infinity.

The answer is then simply I_1 + I_2

If N is not too large, the first integral can be done to the rquired
precision using NIntegrate.

The second integral can be evaluated by noting that the integrand

 1/(x^2(1 + Cos[x]/x^2)) 

can be written as a sum with the general term

 (-1)^n (Cos[x]/x^2)^n /x^2

For integer n, Mathematica can evaluate the indefinite integral over the
general term (although it does not seem to be able to give a general
formula) so the series can be integrated term by term.  Taking enough
terms will give I_2 to arbitrary accuracy.

The trick is to choose N large enough so that you do not need a huge
number of terms in the series that approximates I_2, but not so large that
NIntegrate cannot do the first sub-integral to the desired accuracy. 

If you worked really hard, and extracted the analytic form of the integral
over the general term in I_2, it is just possible you may be able to find
an analytic expression for I_2. If you were really lucky you might then be
able to analytically continue N back to zero, and thus evalute the
integral algebraically.

Richard,
 who after doing this started to wonder why this integral needs to be
worked out to 60 significant figures.

• References:
  • A tough Integral
    • From: bobbym1953@aol.com (bobbym1953)