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Re: How to simplify "Integrate[2 f[x], {x, 0, 1}]/2" to "Integrate[f[x],

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  • Subject: [mg109216] Re: How to simplify "Integrate[2 f[x], {x, 0, 1}]/2" to "Integrate[f[x],
  • From: "David Park" <djmpark at comcast.net>
  • Date: Sat, 17 Apr 2010 06:05:08 -0400 (EDT)

I don't think you are overlooking anything. You could write specific
transformation rules.

But the Presentations package ($50) at my web site has a section called
Student's Integral that allows the manipulation of single variable integrals
before evaluation. It is mainly meant for teaching integration techniques,
but could also be used for derivations in notebooks. You can do breakout of
integrals, and also apply integration techniques such as performing
operations on the integrand, change of variable, integration by parts and
trigonometric substitution. After manipulation, integral can be evaluated
either from a BasicIntegralTable such as students might use, or by using the
built-in Mathematica commands. Or you could create your own integral table.
You might have some complicated integral that takes Mathematica a long time
to evaluate and needs manipulation to a specific form for your use. If you
put this in your own table it will subsequently evaluate very quickly in the
form you desire.

Another nice feature is that the Assumptions are only put in at the time you
evaluate so you can have a nice standard integral display during any
derivation.

In any case, for your example we use only a minimal feature. (Unevaluated
integrals are entered with integrate instead of Integrate. I can't paste the
box form of the results into an email, but just copy and paste into your
notebook.)

Needs["Presentations`Master`"]

1/2 integrate[2 f[x], {x, 0, 1}]
% // BreakoutIntegral

1/2 \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(\(2\ f[x]\)
\[DifferentialD]x\)\)
\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]\(f[x]
\[DifferentialD]x\)\)


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


From: Klaus Engel [mailto:klaus.engel at tiscali.it] 

Dear group,

I tried to simplify an awkward looking integral with "Mathematica 7" 
using its "(Full)Simplify[...]" function. Unfortunately it failed to do 
so, even though I know that this would be possible. I boiled down the 
problem to the following very simple example ("f" is just a generic, 
undefined function): The input

         Integrate[2 f[x], {x, 0, 1}]/2 // FullSimplify

returns just the input

         Integrate[2 f[x], {x, 0, 1}]/2

(same result for "Simplify" instead of "FullSimplify"), i.e., 
Mathematica seems not to be aware that the factor "2" can be canceled 
out. Even worse, the expressions

         TrueQ[Integrate[2 f[x], {x, 0, 1}]/2  == Integrate[f[x], {x, 0, 
1}]]
         SameQ[Integrate[2 f[x], {x, 0, 1}]/2 , Integrate[f[x], {x, 0, 1}]]
         Integrate[2 f[x], {x, 0, 1}]/2  === Integrate[f[x], {x, 0, 1}]

return the (wrong) result "False".

So my question: Is there something I am overlooking, or what is the 
right "Mathematica" way to treat expressions like the one above.

Thanks a lot in advance,

Klaus
------------

Klaus Engel <klaus.engel at tiscali.it>




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