Why does Mathematica give an incorrect answer to this multiple integral?

Let’s learn why does Mathematica give an incorrect answer to this multiple integral. The most accurate or helpful solution is served by Mathematica.

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Why does Mathematica give the wrong answer when integrating?

I integrate Integrate[Exp[I Cos[b - c]] Cos[b], {b, 0, 2 Pi}] Mathematica gives: 2 I Pi BesselJ[1, 1] Which is indepedent on $c$. However, the result is NOT independent on $c$. The numerical value of what Mathematica gives is 0. + 2.76492 I while NIntegrate[Exp[I Cos[b - 0.5]] Cos[b], {b, 0, 2 Pi}] Gives 1.16655*10^-10 + 2.42645 I Certainly not the same. If we multiply the value Mathematica gives with $\cos(0.5)$ the result is the same. What is wrong?

Answer:

Looks like a bug that ran off in the development version of Mathematica. In[1]:= Integrate[Exp[I Cos...

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Jonas Teuwen at Mathematica Mark as irrelevant Undo

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Answer:

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