Re: Integrating x^b*Log[x]^m gives wrong result?

*To*: mathgroup at smc.vnet.net*Subject*: [mg85446] Re: Integrating x^b*Log[x]^m gives wrong result?*From*: "David W.Cantrell" <DWCantrell at sigmaxi.net>*Date*: Mon, 11 Feb 2008 06:03:51 -0500 (EST)*References*: <fomjoj$hqe$1@smc.vnet.net>

KvS <keesvanschaik at gmail.com> wrote: > Dear all, > > I'm running into the following problems with symbolic vs. numerical > integration of the function x^(-3.5)*Log[x]^m: > > In[564]:= > ClearAll["Global`*"]; > f1[m_]:=N[Integrate[x^(-3.5)*Log[x]^m,{x,5,10}]]; > f2[m_]:=NIntegrate[x^(-3.5)*Log[x]^m,{x,5,10}]; > Map[f1,{5,10,25,40}] > Map[f2,{5,10,25,40}] > > Out[567]= {0.145434,4.62609,401145.,-9.30763*10^23} > Out[568]= {0.145434,4.62609,403156.,6.33616*10^10} The discrepancy seen when m is 40 is easily resolved. My first thought was that you should not have used an inexact number, -3.5, in your integrand for the symbolic integration. But only changing that to -35/10, In[10]:= N[Integrate[x^(-35/10)*Log[x]^40, {x, 5, 10}]] Out[10]= -2.50984*10^16 we still see a discrepancy. My second thought was that we also need to ask N for more precision: In[11]:= N[Integrate[x^(-35/10)*Log[x]^40, {x, 5, 10}], 10] Out[11]= 6.336161929*10^10 We see now that the discrepancy is resolved. But it should also be noted that both of my changes are required, that is, just asking N for more precision but leaving the exponent inexact still yields a discrepancy: In[12]:= N[Integrate[x^(-3.5)*Log[x]^40, {x, 5, 10}], 10] Out[12]= -9.30763*10^23 David > Of course the symbolic integration is wrong here since it shouldn't > yield a negative number. If the recursive formula resulting from > partial integration is used, things seem to go wrong as well: > > In[572]:= > f[m_]:=(-1/2.5)*(10^(-2.5)*Log[10]^m-5^(-2.5)*Log[5]^m)+(m/ > 2.5)*f[m-1]; > f[0]=(-1/2.5)*(10^(-2.5)-5^(-2.5)); > Map[f,{5,10,25,40}] > > Out[574]= {0.145434,4.62609,403156.,-2.54037*10^16} > > So the result for m=25 still coincides with the one from NIntegrate, > while Integrate already gives something different; for m=40 the result > is different from both NIntegrate and Integrate (and wrong as it is > negative). If one changes the negative power of x to a positive one, > things seem ok btw. > > Any clues what might be going on here? > > Thanks in advance, Kees > > In[533]:= $Version > Out[533]= 6.0 for Microsoft Windows (32-bit) (April 27, 2007)