Integrate[Sqrt[Cos[t]^n + Sin[t]^n], {t, 0, Pi}] for n=4 and 6
Subject: Integrate[Sqrt[Cos[t]^n + Sin[t]^n], {t, 0, Pi}] for n=4 and 6
From: Alexander Povolotsky
Date: Mon, 7 Sep 2009 17:43:39 -0400
Robert Israel clarified that I was incorrect in feeding Maple's output
into WolframAlpha - I am quoting his reply
---------- Forwarded message ----------
From: Robert Israel
I believe Mathematica's EllipticE[x] is Maple's EllipticE(sqrt(x)).
Maple defines EllipticE(k) = int_0^1 sqrt(1 - k^2 t^2)/sqrt(1 - t^2) dt
so in particular
EllipticE(a*I) = int_0^1 sqrt(1 + a^2 t^2)/sqrt(1 - t^2) dt > 0
for real a.
-------------------------
So Maple's
EllipticE(I)*sqrt(2) for n=4
and
Maple's
EllipticE(sqrt(3)*I) for n=6
are both real and positive.
and hopefully are same values as both from Mathematica and Maxima.
Alex
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