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