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Re: Re: Integrate question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82354] Re: [mg82289] Re: [mg82250] Integrate question
  • From: DrMajorBob <drmajorbob at bigfoot.com>
  • Date: Thu, 18 Oct 2007 04:54:38 -0400 (EDT)
  • References: <200710160728.DAA08846@smc.vnet.net> <31324488.1192621144501.JavaMail.root@m35>
  • Reply-to: drmajorbob at bigfoot.com

I wonder if this is what you mean by "regularized integral":

indef[x_] = Integrate[x/(3 x^2 - 1)^3, x]
indef[1] - indef[0]

-(1/(12 (-1 + 3 x^2)^2))

1/16

But that's meaningless, isn't it, since "indef" isn't continuous on the 
range?

Plot[indef[x], {x, 0, 1}]

Bobby

On Wed, 17 Oct 2007 02:55:32 -0500, Daniel Lichtblau <danl at wolfram.com> 
wrote:

> 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
>
>



-- 

DrMajorBob at bigfoot.com


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