Re: Integrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg74476] Re: Integrate*From*: "David W.Cantrell" <DWCantrell at sigmaxi.net>*Date*: Thu, 22 Mar 2007 01:19:11 -0500 (EST)*References*: <etikok$j7r$1@smc.vnet.net> <etnkdg$gp5$1@smc.vnet.net> <etqnvl$tq$1@smc.vnet.net>

"dimitris" <dimmechan at yahoo.com> wrote: > Hello. > > It is interesting to get responses in this thread because for some > time I believe > I was alone! > > Easily you can get an antiderivative real in the ntegration range: > > (*INs*) > f[x_]=Log[Sin[x]^2]*Tan[x] > integrand = f[x]*dx /. x -> ArcSin[Sqrt[u]] /. dx -> > D[ArcSin[Sqrt[u]], u] > ff=Integrate[integrand, {u, 0, Sin[z]^2}, Assumptions -> 0 < z < Pi] > Simplify[D[ff, z]] /. z -> x > > (*OUTs*) > Log[Sin[x]^2]*Tan[x] > Log[u]/(2*(1 - u)) > (1/12)*(-Pi^2 + 6*PolyLog[2, Cos[z]^2]) Thanks, Dimitris! That's a far simpler result than the one I had given, and it's easily obtained using Mathematica! David > 0 > Log[Sin[x]^2]*Tan[x] > > Plot[ff/.z->x,{x,0,Pi}]; > > Dimitris > > =CF/=C7 David W.Cantrell =DD=E3=F1=E1=F8=E5: > > "dimitris" <dimmechan at yahoo.com> wrote: > > > Hello again! > > > > > > Of course sometimes things work quite unexpectedly! > > > > > > Consider again the integral (no! I am not obsessed with it!) > > > > > > 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,Pi}. > > > > Obsessed or not, it is a curious bug, being platform dependent. > > > > You might be interested in an antiderivative, not directly obtainable > > from Mathematica AFAIK, which is valid over the whole real line: > > > > Letting u = Abs[Cos[x]], > > > > Integrate[Log[Sin[t]^2]*Tan[t], {t, 0, x}] > > > > is > > > > Log[2]^2 - Pi^2/3 + 2 Log[1 + u] Log[(1 + 1/u)/2] + > > 4 Log[Sqrt[2/(1 + u)]] Log[Sqrt[(1 - u)/2]] + 2 PolyLog[2, 1/(1 + u)] > > + PolyLog[2, 2 - 2/(1 + u)] + PolyLog[2, 1 - 2/(1 + u)] > > > > I'm not sure whether the result above could be simplified further or > > not. > > > > David W. Cantrell