       Re: A tough Integral

• To: mathgroup at smc.vnet.net
• Subject: [mg28544] Re: [mg28521] A tough Integral
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 25 Apr 2001 19:21:52 -0400 (EDT)
• References: <200104250530.BAA19306@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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.
>
> Thanks,
> Angela
>
> bobbym1953 at aol.com

60 places?! I wish you luck.

In:= InputForm[nii = NIntegrate[1/(Cos[x]+x^2),  {x,0,Infinity},
WorkingPrecision->200, PrecisionGoal->70, MaxRecursion->12]]

NIntegrate::slwcon:
Numerical integration converging too slowly; suspect one of the
following:
singularity, value of the integration being 0, oscillatory
integrand, or
insufficient WorkingPrecision. If your integrand is oscillatory try
using
the option Method->Oscillatory in NIntegrate.

NIntegrate::ncvb:
NIntegrate failed to converge to prescribed accuracy after 13
recursive bisections in x near x = 18538.4.

Out//InputForm= 1.838017695010507757407163`14.843992684648171

There might be ways to improve on this, but I doubt you'll come close to
the desired precision (the effort tends to increase exponentially in the
number of digits). One possibility might be to show that the integral
from some reasonably small t to infinity differs by no more than
10^-(60) from the integral of 1/x^2 from t to infinity. Then put all
your resources into numerically integrating from 0 to t. But offhand I
do not know if a suitable value for t exists, or, if so, how to find it
(best I can readily get is 10^20, which is too big to be terribly
useful).

Daniel Lichtblau
Wolfram Research

```

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