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Re: Integration error? Integrate[1/(x^3-1)]?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg121708] Re: Integration error? Integrate[1/(x^3-1)]?
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Mon, 26 Sep 2011 20:05:51 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <201109260816.EAA08704@smc.vnet.net>
On 09/26/2011 03:16 AM, Travis Ayres wrote:
> The indefinite integral of 1/(x^3-1) with respect to x.
> In input form, put into Mathematica 8:
> Integrate[1/(x^3 - 1), x]
> Gives result:
> -(ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3]) + (1/3)*Log[1 - x] - (1/6)*
> Log[1 + x + x^2]
>
> Computing online with the Wolfram integrator:
> http://integrals.wolfram.com/index.jsp?expr=1%2F%28x^3-1%29&random=false
> Gives:
> -(ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3]) + Log[-1 + x]/3 - Log[1 + x +
> x^2]/6
>
> Look at the (1/3)*Log[1-x] term.
> Mathematica 8 gives me Log[1-x], the online integrator gives the
> answer Log[x-1].
> The answers are exactly the same in all other terms. I ran across this
> because I was trying the tutorials, and I noticed my answer was
> different than the result in the tutorial even.
>
> Is this an error?
> Thanks all!
>
No. Both are correct antiderivatives (indefinite integrals, primitives).
You can verify this by differentiating and checking that the result, in
each case, is equivalent to the integrand.
The mathematical reason this can happen is that the two terms of
interest differ by a (piecewise) constant. The Mathematica reason is
that the Integrator, for reasons unknown to me, rather insists on using
version 7 of Mathematica. I'll make inquiries about that.
Daniel Lichtblau
Wolfram Research
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