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Integration result depends on variable name / problem with BesselJ Integral representation


Dear all,

the following integral is the integral representation of the bessel function of first kind, second order. But mathematica (8.0.4.0) gives me wrong results, depending on the variable name, it seems:
first with x:
Integrate[Sin[2 x] Exp[I t Cos[x - psi]], {x, 0, 2 \[Pi]}]
(8 I (-t Cos[t] + Sin[t]))/t^2

then x substituted with p:
Integrate[Sin[2 p] Exp[I t Cos[p - psi]], {p, 0, 2 \[Pi]}]
0

How can the result depend on the variable name? There are no values assigned to x or p. The analytically obtained solution should be
2 \[Pi] I^2 BesselJ[2, t] Sin [2 psi]

Regards,
Richard 



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