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Re: Substiuting variables within an integral

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  • Subject: [mg111352] Re: Substiuting variables within an integral
  • From: Alexei Boulbitch <alexei.boulbitch at iee.lu>
  • Date: Wed, 28 Jul 2010 07:25:27 -0400 (EDT)

Hi, Ed,

there is a package "Presentations" written by David Park that contains 
such operators and examples of how to do transformations of that sort.

You may also do it without the package (though less beautiful), if your 
aim is simply to insert a new variable and see what is your integral 
looking like before its actual evaluation. As an example try the one 
below. Assume you are looking for the integral
J=Integrate[func[y], y], where func[y] is a function in question:

rule = {y -> q[x], \[DifferentialD]y -> D[q[x], x]*\[DifferentialD]x};  
(* This is your transformation rule y=q[x], where q is some function *)
underIntegral = func[y] 
\[DifferentialD]y                                            (* This is 
the expression under the integral *) 
underIntegralTransformed = underIntegral /. 
rule                                (* This makes the transformation *)
Integrate[Rest[underIntegralTransformed], 
x]                                      (* This finally calculates the 
integral *)

DifferentialD[y] func[y]

DifferentialD[x] func[q[x]]
\!\(\*SuperscriptBox["q", "\[Prime]",
MultilineFunction->None]\)[x]

\[Integral]func[q[x]]
\!\(\*SuperscriptBox["q", "\[Prime]",
MultilineFunction->None]\)[x] \[DifferentialD]x

Be careful, the use of the operator Rest in the last line  of commands 
is due to the necessity to remove DifferentialD[x], prior to substitute 
the expression into
the operator Integrate. If you use a concrete transform instead y->q[x] 
(such as, for instance, y->x^2), the order of terms in 
underIntegralTransformed may become different, and you will need to use 
another operator to pick up the necessary terms or to remove the 
unnecessary one.

Have fun, Alexei

Hello

I need to do a variable substitution within an integral in several different
ways. I need Mathematica to rearrange the integral with the new variable
definition without solving it.   The resulting integral will be variations
of Euler function.  After seeing the resulting integral, I need Mathematica
to actually solve it for me (that is, Mathematica needs to recognize that it
is an Euler function).  Can this be done?

Many thanks

Ed


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Alexei Boulbitch, Dr. habil.
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