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

*To*: mathgroup at smc.vnet.net*Subject*: [mg85434] Integrating x^b*Log[x]^m gives wrong result?*From*: KvS <keesvanschaik at gmail.com>*Date*: Sun, 10 Feb 2008 05:19:51 -0500 (EST)

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} 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)

**Follow-Ups**:**Re: Integrating x^b*Log[x]^m gives wrong result?***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Integrating x^b*Log ^m gives wrong result?***From:*danl@wolfram.com