       Re: A tough Integral

• To: mathgroup at smc.vnet.net
• Subject: [mg28560] Re: A tough Integral
• From: "Robert Miller" <rmiller at archimedestechnology.com>
• Date: Fri, 27 Apr 2001 03:56:14 -0400 (EDT)
• References: <9c5ndf\$ir1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```The following should do it:
Divide the integration region into lengths of  2 Pi and use
Cos[x]==Cos[x+n*2*Pi] , n an integer to write
Integrate[1/(Cos[x] + x^2), {x, 0, \[Infinity]}] = =
Sum[Integrate[1/(Cos[x] + (x + n*2*Pi)^2), {x, 0, 2  \[Pi]}], {n, 0,
\[Infinity]}]
Interchange the order of summation and integration.
Mathematica can do the sum
sum=Sum[1/(Cos[x] + (x + n*2*Pi)^2), {n, 0, \[Infinity]}]//FullSimplify
It gives the answer in terms of PolyGamma
Then you can NIntegrate sum from 0 to 2 Pi. Note, however, that in the form
in which Mathematica returns the sum, the denominator ->0 at Pi/2 and 3 Pi/2
( the numerator ->0 as well and limit is OK).
So make these endpoints of integration to avoid NIntegrate error messages:
NIntegrate[sum,{x,0 ,Pi/2}]+NIntegrate[sum,{x,Pi/2, 3
Pi/2}]+NIntegrate[sum,{x,3 Pi/2, 2 Pi}]
with WorkingPrecision->80, I get
1.83801769501050385269531439813992926843308653301263139425699189822880091
Robert Miller

"bobbym1953" <bobbym1953 at aol.com> wrote in message
news:9c5ndf\$ir1 at smc.vnet.net...
> 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
>

```

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