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Re: Error in basic integrals

• To: mathgroup at smc.vnet.net
• Subject: [mg7708] Re: Error in basic integrals
• From: gah <gah at math.umd.edu>
• Date: Wed, 2 Jul 1997 14:21:28 -0400 (EDT)
• Organization: University of Maryland, College Park
• Sender: owner-wri-mathgroup at wolfram.com

Sean Ross wrote:
> 
> I hope I have made an error, but I believe I have caught mathematica 3.0 returning
> a dead wrong answer to a basic integral.
> 
> Integrate[1/(1-x^2],x]    returns
> -1/2 Log[-1+x]+1/2 Log[1+x].
> 
> This result is equivalent to ArcCoth[x] which is DEAD WRONG.
> 
> The correct answer is ArcTanh[x] according to CRC handbook etc.  Mathematica
> is retuning the complement to the correct answer.
> 
> Mathematica 2.2.1, by the way, returns the correct answer:  -1/2 Log[x-1] +1/2
> Log[1+x] which is equivalent to ArcTanh[x].
> 
> This is most discouraging.  I have so much code invested in Mma 3.0, that I can't
> change languages in the middle of my dissertation.  The Front end is so buggy,
> but I put up with it because I thought the core numerical language was still OK.
> With something so blatant, how am I supposed to trust the language for things I
> can't easily verify.

The clue is that one answer *is* the complement of the other which means
that they differ by Pi/2, a constant. Antiderivatives or indefinite
integrals are defined only up to additive constants and the two answers
you quote differ only by the  constant Pi/2. If you substitute an upper
limit and subtract the result of substituting a lower limit you will get
the same numerical answer with either formula.
--Garry Helzer