Re: Integral problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg65665] Re: [mg65634] Integral problem*From*: "Carl K. Woll" <carlw at wolfram.com>*Date*: Wed, 12 Apr 2006 06:00:26 -0400 (EDT)*References*: <200604110804.EAA11299@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

ivan.svaljek at gmail.com wrote: > I've tried running an integral expression through mathematica 5.0 and > came up with this: > > http://aspspider.net/isvaljek/mathematica/index.html > > First two are from mathematica, and the last one is from a site running > web mathematica. The first set of input/output was: In[1]:= Integrate[x^3/(x^8-2),x] Out[1]= (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: In[2]:= D[Log[x-1],x]//Simplify D[Log[1-x],x]//Simplify Out[2]= 1/(-1+x) Out[3]= 1/(-1+x) 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

**References**:**Integral problem***From:*ivan.svaljek@gmail.com