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[16]:= 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[16]//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

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