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Re: simplifying definite vs indefinite integrals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90661] Re: simplifying definite vs indefinite integrals
  • From: "David Park" <djmpark at comcast.net>
  • Date: Fri, 18 Jul 2008 04:01:54 -0400 (EDT)
  • References: <g5n464$sbm$1@smc.vnet.net>

Rik,

It is strange that Mathematica factors out a[x] in one case and not in the 
other.

In any case, the Student's Integral section of the Presentations package 
allows a user to manipulate an unevaluated integral (operating on the 
integrand, doing a change of variable, using integration by parts, using a 
trigonometric substitution, and breaking out sums and nondependent factors) 
before submitting the integral either to an integral table or to the regular 
Mathematica Integrate. It uses integrate with a small i. So:

Needs["Presentations`Master`"]

integrate[a[x] b[y], {y, s, t}]
% // BreakoutIntegral

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(\(a[x]\ b[y]\) 
\[DifferentialD]y\)\)
a[x] \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(b[y] 
\[DifferentialD]y\)\)

Or if you want to do your 'hand factoring' method you could use:

integrate[a[x] b[y], {y, s, t}]
a[x] % // OperateIntegrand[#/a[x] &]

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(\(a[x]\ b[y]\) 
\[DifferentialD]y\)\)
a[x] \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(b[y] 
\[DifferentialD]y\)\)


-- 
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/


<rikblok at gmail.com> wrote in message news:g5n464$sbm$1 at smc.vnet.net...
> Hi Mathematica gurus (& sorry if this is a dupe post),
>
> I'm new to Mathematica and I was surprised to see that it handles
> definite versus indefinite integrals differently. For example:
>
> In[1]:= indef = Integrate[a[x] b[y], y]
>
> Out[1]= a[x] \[Integral]b[y] \[DifferentialD]y
>
> Nice.  a[x] gets pulled out of the integral.
>
> In[2]:= def = Integrate[a[x] b[y], {y, s, t}]
>
> Out[2]= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(\(a[x]\ b[
>    y]\) \[DifferentialD]y\)\)
>
> But not for the definite integral.  Why?  And how can I make it factor
> out?
>
> In[3]:= Collect[def, a[x]]
>
> Out[3]= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(\(a[x]\ b[
>    y]\) \[DifferentialD]y\)\)
>
> doesn't work. Nor does
>
> In[4]:= Simplify[def]
>
> Out[4]= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(\(a[x]\ b[
>    y]\) \[DifferentialD]y\)\)
>
> I can't even remove a[x] manually:
>
> In[5]:= FullSimplify[def/a[x]]
>
> Out[5]= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(s\), \(t\)]\(\(a[x]\ b[
>    y]\) \[DifferentialD]y\)\)/a[x]
>
> Suggestions?  Thanks for your help!
>
> Rik
> 



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