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
>
>
> Integrate[f,{x,0,Infinity}]
>
>
> -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.

>
> 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

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?

```

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