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