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Re: Integral confusion

  • To: mathgroup at
  • Subject: [mg107297] Re: Integral confusion
  • From: "Nasser M. Abbasi" <nma at>
  • Date: Mon, 8 Feb 2010 03:36:16 -0500 (EST)
  • References: <hkm7d8$os0$>

"Jon Joseph" <josco.jon at> wrote in message 
news:hkm7d8$os0$1 at
> All: Is this integral wrong? If not could someone explain the minus sign
> inside the log?
> Integrate[1/(x + 1) - 1/(x + 6), x] // Simplify
> log(-2 (x + 1)) - log(2 (x + 6))
> Thanks, Jon.=

Well, lets see:

log(-2 (x + 1)) - log(2 (x + 6))
          = log(-2)+log(1+x) -log(2)-log(x+6)
          = log(-1)+log(2)+log(1+x)-log(2)-log(x+6)
          = log(-1)+ log(1+x) - log(x+6)

but log(-1) = Sqrt[-1]*Pi

so result is

         Sqrt[-1]*Pi + log(1+x) - log(x+6)

But we all know that the result should be

         log(1+x) - log(x+6)

So, an extra term, Sqrt[-1]*Pi term pops up. But this term is a constant, so 
its derivative is zero, i.e. a constant of integration.


D[log(-2 (x + 1)) - log(2 (x + 6)),x] will give back

               1/(x + 1) - 1/(x + 6)

So, in theory, the answer given by Mathematica is NOT wrong.

But Mathematica does (normally?) return results for indefinite integrals 
without an explicit constant of integration. So I am not sure why it does in 
this case, and if it it does, why did not pick this constant? Why not C[1] 
as it does for DSolve[]?

So, if I have to guess, I'd say this result is at least very weired, but 
mathematically it is not wrong?


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