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MathGroup Archive 2006

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Re: Integral problem

  • To: mathgroup at
  • Subject: [mg65665] Re: [mg65634] Integral problem
  • From: "Carl K. Woll" <carlw at>
  • Date: Wed, 12 Apr 2006 06:00:26 -0400 (EDT)
  • References: <>
  • Sender: owner-wri-mathgroup at

ivan.svaljek at wrote:
> I've tried running an integral expression through mathematica 5.0 and
> came up with this:
> First two are from mathematica, and the last one is from a site running
> web mathematica.

The first set of input/output was:


(Log[Sqrt[2] - x^4] - Log[Sqrt[2] + x^4])/(8*Sqrt[2])

The second output (webMathematica) gave

(Log[x^4 - Sqrt[2]] - Log[Sqrt[2] + x^4])/(8*Sqrt[2])

I guess the question is why the argument of the first log changes sign? 
Note that (up to branch issues)

Log[Sqrt[2]-x^4] == Log[(-1)(x^4-Sqrt[2])] == Log[-1]+Log[x^4-Sqrt[2]]

So, the difference between the two expressions is a constant, albeit a 
complex constant (Log[-1]==Pi I). Since an indefinite integral is only 
defined up to a constant, both answers are correct.

A simpler manifestation of this issue:




Both Log[x-1] and Log[1-x] have a derivative equal to 1/(x-1), so they 
must both be indefinite integrals of 1/(x-1).

Carl Woll
Wolfram Research

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