Re: A wrong definite integral in 5.0?

• To: mathgroup at smc.vnet.net
• Subject: [mg77319] Re: A wrong definite integral in 5.0?
• From: dh <dh at metrohm.ch>
• Date: Wed, 6 Jun 2007 07:32:06 -0400 (EDT)
• References: <f43hau\$301\$1@smc.vnet.net>

```
Hi,

you will remember from calculus that Log is a multivalued function:

principal value + 2Pi n I. Therefore, the results are correct (but not

as consistent as you would like it).

Hint: in version 6 you get -1 for both integrals.

hope this help, Daniel

bolud-el-kotur wrote:

> 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.

>

>

```

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