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?