Re: Bug in Integrate in Version 5.1?

*To*: mathgroup at smc.vnet.net*Subject*: [mg56814] Re: [mg56737] Bug in Integrate in Version 5.1?*From*: yehuda ben-shimol <bsyehuda at gmail.com>*Date*: Fri, 6 May 2005 03:01:28 -0400 (EDT)*References*: <200505051001.GAA21888@smc.vnet.net>*Reply-to*: yehuda ben-shimol <bsyehuda at gmail.com>*Sender*: owner-wri-mathgroup at wolfram.com

There is more peculiar result there: Try the indefinite integral a=Integrate[-3 (x^2 )Log[1 - Exp[-x]], x] to get -3*(x^4/12 + (1/3)*x^3*Log[1 - E^(-x)] + (1/3)*((-x^3)*Log[1 - E^x] - 3*x^2*PolyLog[2, E^x] + 6*x*PolyLog[3, E^x] - 6*PolyLog[4, E^x])) this expression does not have value in both x=0 and x-> infinity but it does converge to a limit for both So Limit[a,x->Infinity]-Limit[a,x->0] does give the exact resut (i.e., Pi^4/15) I cannot figure out why using the definite integral does not return the true value. yehuda On 5/5/05, A.Reischl at gmail.com <A.Reischl at gmail.com> wrote: > Hello, > > Integrate gives the following answer for this integral: > > a = Integrate[x^3 /(Exp[x] - 1), {x, 0, Infinity}] > N[a] > > Out[1]= Pi^4/15 > Out[2]= 6.49394 > > which I think is correct. > This integral, which should be the same ( by partial integration), > gives: > b = Integrate[-3 x^2 Log[1 - Exp[-x]], {x, 0, Infinity}] > N[b] > > Out[3]= (11*Pi^4)/60 > Out[4]= 17.8583 > > while numerical integration gives: > NIntegrate[-3x^2 Log[1 - Exp[-x]], {x, 0, Infinity}] > Out[5]= 6.49394 > > This is done with version 5.1. > > Version 4.2 gives > c=Integrate[-3*x^2*Log[1 - Exp[-x]], {x, 0, Infinity}] > N[c] > > Out[1]= Pi^4/15 > Out[2]= 6.49394 > > (Remarkably version 4.2. complaints: "Series::esss: Essential > singularity > encountered in ..." while calculating the correct result. ) > > So the result in version 5.1. looks wrong. > Or did I make a mistake? > > Cheers > Alexander > >

**References**:**Bug in Integrate in Version 5.1?***From:*A.Reischl@gmail.com