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Re: Inconsistent behaviour of Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112476] Re: Inconsistent behaviour of Integrate
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Thu, 16 Sep 2010 06:03:01 -0400 (EDT)

$Version

"7.0 for Mac OS X x86 (64-bit) (February 19, 2009)"

I see the same behavior. In the simplification process it appears to fall through a trap door (i.e., transform rule that it only knows in one direction). At the expense of slow calculation, you can get the first result consistently by making it forget.

Module[{},
 ClearSystemCache[];
 Integrate[
  Sqrt[(x - 1/2)^2 + (y - 1/2)^2],
  {x, 0, 1}, {y, 0, 1}]]

(1/6)*(Sqrt[2] + ArcSinh[1])


Bob Hanlon

---- Andreas Maier <andimai at web.de> wrote: 

=============
Hello,

I'm using Mathematica 7.0.1.0 on Linux x86 (64bit). I have a notebook
file, where I integrate the same integral twice:

In[1]:= Integrate[Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0,
1}]
Out[1]= 1/6 (Sqrt[2] + ArcSinh[1])

In[2]:= Integrate[Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0,
1}]
Out[2]= 1/24 (4 Sqrt[2] + Log[17 + 12 Sqrt[2]])

As you can see from the output, integrating the same integral a second
time gives a different result. If I integrate the same integral a
third and a fourth time I always get the second result again. Only if
I restart the mathematica kernel, I get the first result again.
The results are equivalent, since

Log[17 + 12 Sqrt[2]] = Log[(1 + Sqrt[2])^4] = 4* Log[(1 + Sqrt[2]) = 4* ArcSinh[1]

but somehow Mathematica seems to be able to do this simplification
only once. Is this inconsistent behaviour a bug? Is there a
possibility to give mathematica a hint, so that he always find the
first solution 1/6 (Sqrt[2] + ArcSinh[1]) to the integral?
From

In[3]:= Expand[(1 + Sqrt[2])^4]
Out[3]= 17 + 12 Sqrt[2]

In[4]:= Factor[%]
Out[4]= 17 + 12 Sqrt[2]

I also figured that Mathematica doesn't seem to be able to factorize
an expression like 17 + 12 Sqrt[2] into (1 + Sqrt[2])^4. Is this a
known problem? Or should I use a different command to find this
factorization?

Sincerely,
Andreas Maier




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