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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
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