Re: Can Mathematica NIntegrate a Log-type singularity?

*To*: mathgroup at smc.vnet.net*Subject*: [mg92662] Re: [mg92642] Can Mathematica NIntegrate a Log-type singularity?*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Fri, 10 Oct 2008 04:32:07 -0400 (EDT)*References*: <200810091036.GAA24061@smc.vnet.net>

On 9 Oct 2008, at 19:36, Aaron Fude wrote: > Hi, > > I would like to evaluate the following: > > NIntegrate[alpha^2 Log[2 Cos[alpha/2]]^2, { alpha, -Pi, Pi}] > > How do I help Mathematica deal with the LogSquared type singularity at > either end of the interval. If I try it straight, Mathematica > complains and gives a wrong answer. > > Please note, that Mathematica has absolutely no problem Integrating or > NIntegrating the function > > Log[Cos[alpha/2]]^2 > > from -Pi to Pi, each time giving the correct answer, but the multiple > of alpha^2, throws it off. > > Many thanks in advance, > > Aaron. > > PS: By the way, I'm pretty sure that that integral must be some > rational number times Pi^5. > Which version of Mathematica? Here with version 6.03 I get: a = Chop[NIntegrate[alpha^2*Log[2*Cos[alpha/2]]^2, {alpha, -Pi, Pi}, WorkingPrecision -> 30]] 37.402405918201066509890604560864352814270789949531176\ 7786391`30. and then: RootApproximant[a/Pi^5] 11/90 Andrzej Kozlowski

**References**:**Can Mathematica NIntegrate a Log-type singularity?***From:*Aaron Fude <aaronfude@gmail.com>