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

• To: mathgroup at smc.vnet.net
• Subject: [mg85481] Re: [mg85434] Integrating x^b*Log[x]^m gives wrong result?
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Mon, 11 Feb 2008 06:22:19 -0500 (EST)

```It is a precision issue

Clear[x, m];

f1[m_] := Integrate[x^(-7/2)*Log[x]^m, {x, 5, 10}];

f2[m_] := NIntegrate[x^(-3.5)*Log[x]^m, {x, 5, 10}];

f3[m_] := Integrate[x^(-3.5)*Log[x]^m, {x, 5, 10}];

f4[m_] := Integrate[x^(-3.5`25)*Log[x]^m, {x, 5, 10}];

v = {25, 40};

Machine precision:

N[f1 /@ v]

{403155.55804573477, -4.183061773182904*^16}

Extended precision:

N[f1 /@ v, 20]

{
403156.30667862031262341904898345961`19.99999999999\
9996,
6.3361619293717813936101205761`20.000000000000007*^\
10}

Machine precision:

f1 /@ N[v]

{403156.25, 0.}

Extended precision:

f1 /@ N[v, 23]

{403156.306678620312623419`13.159208013405054,
6.3361619293717814121`1.6036615637884584*^10}

Numerical integration:

f2 /@ v

{403156.306678621, 6.3361619293717926*^10}

Machine precision:

f3 /@ v

{401853.125, -5.48409663189581*^23}

Extended precision:

f4 /@ v

{403156.3066786203126239621`14.540465311148813,
6.335827093390950705`2.984895910513547*^10}

Bob Hanlon

---- 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}
>
> 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?
>