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MathGroup Archive 2007

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Re: Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74426] Re: Integrate
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Wed, 21 Mar 2007 02:45:48 -0500 (EST)
  • References: <etikok$j7r$1@smc.vnet.net><etnkdg$gp5$1@smc.vnet.net>

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])
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



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