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Re: Change integral variables
*To*: mathgroup at smc.vnet.net
*Subject*: [mg88217] Re: Change integral variables
*From*: "David Park" <djmpark at comcast.net>
*Date*: Mon, 28 Apr 2008 04:41:34 -0400 (EDT)
*References*: <fv1fep$es1$1@smc.vnet.net>
The Presentations package at my web site below ($50) has a Student's
Integral section that allows students to display and manipulate integrals
using basic integration techniques. It also has a BasicIntegralTable such as
students might use. So it is possible to do many integrals completely
bypassing the Mathematica Integrate command, or it is possible to manipulate
an integral to a starting form and then hand it over to Integrate.
Here is your integral done with Student's Integral. The integrate command is
the inert form of Integrate that you were asking for. We first use a change
of variable, then do an integration by parts, use the BasicIntegralTable,
and then resubstitute the original variable. The displays would contain the
normal box form for an integral but I can only show the plain text form in
the posting. I usually do such calculations all in one cell, building up the
steps and reevaluating.
integrate[Sin[Sqrt[x]], x]
% // ChangeIntegralVariable[tsub = t -> Sqrt[x], x]
% // IntegrateByParts[t, t]
% + C // UseIntegralTable[BasicIntegralTable]
% /. tsub
The successive outputs are: (You should be able to paste these into a
notebook.)
\[Integral]Sin[Sqrt[x]]\[DifferentialD]x
\[Integral]2 t Sin[t]\[DifferentialD]t
-2 t Cos[t]+2 \[Integral]Cos[t]\[DifferentialD]t
C-2 t Cos[t]+2 Sin[t]
C-2 Sqrt[x] Cos[Sqrt[x]]+2 Sin[Sqrt[x]]
And that checks with the simple use of the Integrate command.
In addition, in the IntegrateByParts step, there was generated a Panel
display giving the integration by parts information. It looked like the
following:
u = t, du = 1 dt
dv = 2 Sin[t] dt, v = -2 Cos[t]
Used BasicIntegralTable successfully.
In the IntegrateByParts command there is a choice of using an integral
table, or using the Mathematica Integrate command.
In addition to these commands there are commands: OperateIntegrand to
perform algebraic operations on the integrand, TrigonometricSubstitute for
the standard trigonometric substitutions (which also generates a triangle
plot) and BreakoutIntegral for breaking out sums and constants from integral
expressions. The routines will also handle definite integrals and will
display intermediate forms with a LimitsBracket.
Although this is basically for students, it can also be useful in writing
tutorials or research papers where one may wish to show some basic
techniques in handling integrals. Sometimes a Mathematica Integrate will be
very slow and return the answer in a form that has to be further manipulated
to a desired form. A special integral table could be constructed to handle
such cases.
--
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/
"Budaoy" <yaomengliang at gmail.com> wrote in message
news:fv1fep$es1$1 at smc.vnet.net...
> In Mathematica, how can I change integral variables? For example in
> integration:
>
> Integrate[Sin[Sqrt[x]], x] if I want to use t^2=x to instead x in the
> integral, how can I achieve this?
>
> PS: Does Mathematica have a inertial form of some symbolic command,
> for instance, the above integration, if I only want an integration
> form but not an answer, what can I do?
>
> Thanks.
>
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