symbolic integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg74213] symbolic integration*From*: "dimitris" <dimmechan at yahoo.com>*Date*: Wed, 14 Mar 2007 03:55:33 -0500 (EST)

I am reading about Symbolic Integration from Manuel Bronstein's Symbolic Integration 1 and some references therein. >From the articles: "Indefinite and Definite Integration" by Kelly Roach (1992) "The evaluation of Bessel functions via G-function identities" by Victor Adamchik (1995) "Definite Integration in Mathematica 3.0" by the same author "Symbolic Definite Integration" by Daniel Lichtblau I was able to figure out a lot of things on how Mathematica determines indefinite and definite integrals. What I don't understand (and I can't find any clear reference anywhere) is how Mathematica having evaluated an antiderivative "searches for" and "figures out" possible singulaties in the integration range. For example here (*non-integrable singularity at x=1/2*) Block[{Message}, Integrate[1/(x - 1/2), {x, 0, 1}]] Infinity and here... Integrate[1/Sqrt[x], {x, 0, 1}] (*integrable singularity*) 2 Also how about here Integrate[1/(2 + Cos[x]), {x, 0, 2*Pi}] (2*Pi)/Sqrt[3] where in this example the integrand is a continuus function of x but the indefinite integral return by Mathematica has a finite discontinuity at x=Pi Integrate[1/(2 + Cos[x]), x] Show@Block[{$DisplayFunction=Identity},Plot[%, {x,#[[1]],#[[2]]}]&/ @Partition[Range[0,2Pi,Pi],2,1]] (2*ArcTan[Tan[x/2]/Sqrt[3]])/Sqrt[3] Next Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}] Integrate::idiv : Integral of Log[Sin[x]^2]*Tan[x] does not converge on \ {x,0,=CF=80}. Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}] Of course the integral is convergent. Tr@(NIntegrate[Log[Sin[x]^2]* Tan[x],{x,#[[1]],#[[2]]},WorkingPrecision\[Rule]40]&/ @Partition[\ Range[0,Pi,Pi/2],2,1]) 0``29.90647836759534 Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi/2, Pi}] 0 Plot[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}]; f = (FullSimplify[#1, 0 <= x <= Pi] & ) [Integrate[Log[Sin[x]^2]*Tan[x], x]] (Plot3D[Evaluate[#1[f /. x -> x + I*y]], {x, 0.01, Pi}, {y, 0.01, Pi}, PlotPoints -> 50] & ) /@ {Re, Im}; Any insight will be greatly appreciate.