Re: Integration bug? Integrate[Sin[2 t]^2 Cos[Cos[t - p]], {t, 0, 2 Pi}]

*To*: mathgroup at smc.vnet.net*Subject*: [mg126342] Re: Integration bug? Integrate[Sin[2 t]^2 Cos[Cos[t - p]], {t, 0, 2 Pi}]*From*: "jf.alcover" <jf.alcover at gmail.com>*Date*: Thu, 3 May 2012 04:34:14 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jndti3$jik$1@smc.vnet.net>

Le vendredi 27 avril 2012 12:48:35 UTC+2, sykesy a =E9crit : > Hi all, > > I am trying to compute the following integral > > Integrate[Sin[2 t]^2 Cos[Cos[t - p]], {t, 0, 2 Pi}, Assumptions->p>0] > > for which Mathematica gives me the answer, > > 8 Pi (BesselJ[2,1]-BesselJ[3,1]) > > I was confused as to why the integral did not depend on p. > > I used the following numerical approach to the problem, > > Plot[NIntegrate[Sin[2 t]^2 Cos[Cos[t - p]], {t, 0, 2 Pi}], {p, 0, 2 > Pi}] > > And found that the integral did depend on p. > > Is there something I am missing here? For a long time I thought I must > be doing something very obvious wrong, but I can't see what it would > be. Can anybody help me with this issue? > > best wishes, > Andrew Sykes Hello, Here is an "empirical" closed form of the integral: 8*Pi*(BesselJ[2, 1] - BesselJ[3, 1])*Cos[2*p]^2 + Pi*(40*BesselJ[1, 1]-22*BesselJ[0, 1])*Sin[2*p]^2 hth