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MathGroup Archive 2007

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Re: Integrate (fix a mistake)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74231] Re: Integrate (fix a mistake)
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Thu, 15 Mar 2007 04:57:29 -0500 (EST)
  • References: <esls78$q0v$1@smc.vnet.net><et5o76$jor$1@smc.vnet.net>

Let me mention one mistake (and only one, I hope!)  in my message:

integrand = f[x]*dx /. x -> ArcSin[Sqrt[u]] /. dx ->
D[ArcSin[Sqrt[u]], u]
Integrate[integrand, {u, 0, Sin[z]^2}, Assumptions -> 0 < z < Pi]
Plot[%, {z, 0, Pi}]
(%% /. z -> Pi) - (%% /. z -> 0)
Simplify[D[%%%, z]] /. z -> x

Log[u]/(2*(1 - u))
(1/12)*(-Pi^2 + 6*PolyLog[2, Cos[z]^2])
Graphics[]
0
Log[Sin[x]^2]*Tan[x]

Cheers
Dimitris


=CF/=C7 Michael Weyrauch =DD=E3=F1=E1=F8=E5:
> Hello,
>
>   Dimitris, this is a marvelous solution to my problem. I really apprecia=
te
> your help. I will now see if I can solve all my other (similar) integrals=
 using the same trick.
> Timing is not really the big issue if I get results in a reasonable amoun=
t of time.
>
> Also the references you cited are quite interesting, because they give so=
me insight
> what might go on inside Mathematica concerning integration.
>
> I also agree with you that it is my task as a programmer to "help" Mathem=
atica
> and guide it into the direction I want it to go. Sometimes this is an "ar=
t"...
>
> Nevertheless, in the present situation I do not really understand why Mat=
hematica wants
> me to do that rather trivial variable transformation, which is at the hea=
rt of your solution.
> The integrand is still a rather complicated rational function of the same=
 order. The form
> of the integrand did not really change
> substantially as it is the case with some other ingenious substitutions o=
ne uses in order to
> do some complicated looking integrals "by hand".
>
> I think the fact that we are forced to such tricks shows that the Mathema=
tica integrator
> is still a bit "immature" in special cases, as also the very interesting =
article by D. Lichtblau,
> which you cite, seems to indicate. So all this is probably perfectly know=
n to the
> Mathematica devellopers. And I hope the next Mathematica version has all =
this "ironed out"??
>
> Many thanks again,  Michael



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