       A wrong definite integral in 5.0?

• To: mathgroup at smc.vnet.net
• Subject: [mg77201] A wrong definite integral in 5.0?
• From: bolud-el-kotur <ke8eqet at hotmail.com>
• Date: Tue, 5 Jun 2007 06:56:05 -0400 (EDT)

```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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