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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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