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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*: Wed, 2 May 2012 05:46:29 -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
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