Re: Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg74401] Re: Integrate
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Mon, 19 Mar 2007 22:10:12 -0500 (EST)
- References: <etikok$j7r$1@smc.vnet.net>
"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