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