Re: Trouble with Integrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg39272] Re: Trouble with Integrate*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Fri, 7 Feb 2003 03:07:45 -0500 (EST)*References*: <b1t66t$944$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

"Marko Vojinovic" <vojinovi at panet.co.yu> wrote: > Consider the function: > > f = Sqrt[1+x^4] -x^2 > > Upon asking to > > Integrate[f,{x,0,Infinity}] > > Mathematica 4.0 answers: > > -Infinity > > which is not correct. Alas, so does my Version 4.2.0.0 for Windows. > However, > > NIntegrate[f,{x,0,Infinity}] > > gives the correct (numerical) answer: > 1.23605 > > The correct (analytical, i.e.. exact) answer to the integral is: > > Gamma[1/4] Gamma[1/4] / 6 Sqrt[Pi] > > which can be obtained after some paperwork. Using Integrate[f,{x,0,a}], I got an answer involving EllipticF. Its limit as a -> Infinity is correct, although Mathematica cannot find that limit. > However, if I ask > > Integrate[1/(Sqrt[1+x^4] + x^2),{x,0,Infinity}] > > (this integrand is equivalent to f) one gets a complicated answer in > terms of EllipticF. Strange. My later version doesn't give an answer, merely rewriting the problem in 2D form. > Meanwhile, when I ask Mathematica 3.0 the same set of > questions, I get correct answers, Aargh! Now I wish that I'd left version 3 on my computer! Please report these problems to the proper authorities, if you haven't already done so. David > and analytical integration gives answer > in terms of Gamma. Two questions: > > 1) Why does version 4.0 give so fairly incorrect result "-Infinity" for > the first integral? > 2) How can I 'switch off' the use of elliptic functions and/or 'force' > Mathematica to use Gamma?