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Re: Integrate question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82289] Re: [mg82250] Integrate question
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 17 Oct 2007 03:55:32 -0400 (EDT)
  • References: <200710160728.DAA08846@smc.vnet.net>

Oskar Itzinger wrote:
> Mathematica 5.2 under IRIX complains that
> 
> Integrate[x/(3 x^2 - 1)^3,{x,0,1}]
> 
> doesn't converge on [0,1].
> 
> However, Mathematica 2.1 under Windows gives the corrrect answer, (1/16).
> 
> When did Mathematica lose the ability to do said integral?
> 
> Thanks.

There is a nonintegrable singularity at 1/Sqrt[3]. You can check this 
via Series.

In[6]:= Series[x/(3*x^2 - 1)^3, {x,1/Sqrt[3],2}]

                  1                          1                     1
Out[6]= -------------------- - ---------------------------- + ---------- -
                  1         3                    1         2   32 Sqrt[3]
         72 (-(-------) + x)    48 Sqrt[3] (-(-------) + x)
               Sqrt[3]                        Sqrt[3]

              1                            1         2
      5 (-(-------) + x)   9 Sqrt[3] (-(-------) + x)
           Sqrt[3]                      Sqrt[3]                 1         3
 >    ------------------ + --------------------------- + O[-(-------) + x]
             128                       256                   Sqrt[3]

To get a regularized integral you can use GenerateConditions->False.

In[7]:= Integrate[x/(3*x^2 - 1)^3, {x,0,1}, GenerateConditions->False]

         1
Out[7]= --
         16

I realize this is an overloaded option and expect that in some future 
version there will be a cleaner way to request regularized results of 
definite integration.


Daniel Lichtblau
Wolfram Research


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