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MathGroup Archive 2003

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Trouble with Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39264] Trouble with Integrate
  • From: "Marko Vojinovic" <vojinovi at panet.co.yu>
  • Date: Thu, 6 Feb 2003 03:08:30 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

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. 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. 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. Meanwhile, when I ask Mathematica 3.0 the same set of
questions, I get correct answers, 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?

Thanks,
                             Marko









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