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Indefinite integration problem on ArcTanh(f(Cos))

• To: mathgroup at smc.vnet.net
• Subject: [mg69933] Indefinite integration problem on ArcTanh(f(Cos))
• From: "ANDREW PALFREYMAN" <andrew_ppp at hotmail.com>
• Date: Wed, 27 Sep 2006 06:06:26 -0400 (EDT)

Hi.
I'm integrating Sin[t] / (1 + k Cos[2t]) , over 0 < k < 1, using Mathematica 4.1. I'm plotting spot values k=0.1, 0.8, 0.95 over  0 
< t < 10.

This represents the acceleration of a charged particle in an alternating electric field, where k represents a particular modulation 
of its effective inertia. Although the velocity plots (the integral dt) look fine (symmetrical about zero), the plots for position 
(integrating velocity dt) are clearly wrong; I get a massive expression (about 20 lines!) which will not differentiate successfully 
back to the original velocity expression, and they are decidedly asymmetrical about zero. The position plot ought to look like a 
sinewave as k->0, but there's no resemblance.

Andrew Palfreyman