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

*To*: mathgroup at smc.vnet.net*Subject*: [mg85477] Re: [mg85434] Integrating x^b*Log[x]^m gives wrong result?*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Mon, 11 Feb 2008 06:20:13 -0500 (EST)*References*: <200802101019.FAA17904@smc.vnet.net>

On 10 Feb 2008, at 11:19, KvS 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} > > 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) The only thing that is wrong is that you are using machine precision arithmetic with Integrate and run into numerical instability. Observe: a = Integrate[x^(-7/2)*Log[x]^40, {x, 5, 10}]; with ten digits of precision: N[a, 10] 6.336161929*10^10 with machine precision: N[a] -4.183061773182904*^16 The moral is obvious. Its risky to use machine precision unless you are able to convince yourself that the formula you are evaluating is not numerically ill conditioned. Andrzej Kozlowski > >

**References**:**Integrating x^b*Log[x]^m gives wrong result?***From:*KvS <keesvanschaik@gmail.com>