Re: A wrong definite integral in 5.0? (2nd response)
- To: mathgroup at smc.vnet.net
- Subject: [mg77280] Re: A wrong definite integral in 5.0? (2nd response)
- From: dimitris <dimmechan at yahoo.com>
- Date: Wed, 6 Jun 2007 07:11:49 -0400 (EDT)
- References: <f43hau$301$1@smc.vnet.net>
As a second workaround i suggest you trying in Mathamatica 5.0
the following
In[153]:=
Integrate[Exp[(-x)*s]*Log[1 - 4*x*(1 - x)], {x, 0, 1},
GenerateConditions -> False]
Out[153]=
-((4*SinhIntegral[s/2])/(E^(s/2)*s))
In[154]:=
Limit[%, s -> 0]
Out[154]=
-2
Does it work?
Dimitris
/ bolud-el-kotur :
> I get this result in version 5.0,
>
> >Integrate[Log[1 - 4 x(1 - x)], {x, 0, 1}]
> >-2 + I Pi
>
> and the same thing if I "declare" the singularity with {x,0,1/2,1}.
>
> Another way to look at the problem is computing,
>
> >Integrate[Log[1 - 4 x(1 - x)], {x, 0, 1/2}]
> >-1
>
> and
>
> >Integrate[Log[1 - 4 x(1 - x)], {x, 1/2, 1}]
> >-1 + I Pi
>
> Since the integrand is symmetric about x=1/2, the result should have
> been the same one (-1) in both cases, and the integral over [0,1]
> should yield -2.
>
> A numerical approach,
>
> >NIntegrate[Log[1 - 4 x(1 - x)], {x, 0, 1/2, 1},
> MaxRecursion -> 100, SingularityDepth -> 20]
> >-1.9999997086422834`
>
> gives the correct result, within the numerical accuracy required.