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customizing Integrate with Unprotect

• To: mathgroup at smc.vnet.net
• Subject: [mg49273] customizing Integrate with Unprotect
• From: "S. Narayan" <narayans at btv.ibm.com>
• Date: Sat, 10 Jul 2004 02:48:51 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

For some custom function F, I used Unprotect to "teach" Mathematica the 
integration (which involves a recursive definition) I am unable to use 
this integration in more than one dimension.

Here is the example (in text, I have the mathematica session at the end 
of the post)

Unprotect[Integrate];
Clear[Integrate];
Integrate[x_^n_ * F[a_ + b_* x_], x_] :=
(n+1)a^n *Integrate[x^(n-1) * F[a + b*x], x] + (n-1)*b^n;
Integrate[x_ * F[a_ + b_*x_], x_] := 2* a * F[a/b];
Integrate[F[a_ + b_*x_], x_] := F[a/b];

I require that n be a whole number and a, b real numbers.
I tried a couple of examples in one dimension and it works correctly:

Integrate[x*x*F[a + b*x], x]

 > b^2 + 6 a^3 F(a/b);

When I try it in multiple dimensions, I don't get the right answer:

Integrate[y*F[a + b*x + c*y], y]
 > 2*(a + b*x)* F[(a + b*x)/ c]
Integrate[%, y]
 > 2Integrate[(a + b*x)*F[(a + b*x)/c], x]

Note that it does not apply the integration form from above, correctly.

Is there anyway I can get this integration performed in Mathematica ? 
(My problem is I have to do this nested integration in more than 2 
dimensions and cannot do it manually)

Thanks,
- narayan




(* Here is the mathematica notebook  *)


\!\(\(Unprotect[Integrate];\)\[IndentingNewLine]
   \(Clear[Integrate];\)\[IndentingNewLine]
   \(Integrate[x_\^n_*F[a_\  + \ b_*x_], x_] := \((
     n + 1)\)*a\^n*Integrate[x\^\(n - 1\)*F[a\  + \ b*x],
      x]\  + \ \((n - 1)\)*b\^n;\)\n
   \(Integrate[x_*F[a_\  + \ b_*x_], x_] := 2*a*F[a\/
       b]\ ;\)\[IndentingNewLine]
   \(Integrate[F[a_\  + \ b_*x_], x_] := \ F[a\/b];\)\[IndentingNewLine] 
(*\ \
run\ some\ examples\ *) \[IndentingNewLine]
   Integrate[F[a\  + \ b*x], \ x]\[IndentingNewLine]
   Integrate[x*F[a\ \  + \ b*x], \ x]\[IndentingNewLine]
   Integrate[x*x*F[a\  + \ b*x], \
     x]\[IndentingNewLine] (*\ now\ try\ two\ dimensions\ *) \
\[IndentingNewLine]
   Integrate[y*F[a\  + \ b*x\  + \ c*y], \ y]\[IndentingNewLine]
   Integrate[%, \ x]\)

• Follow-Ups:
  • Re: customizing Integrate with Unprotect
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>