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How can I calculate the following integral (wisely) in Mathematica?

$\qquad \int^{2\pi}_0 \int^{\pi}_0 \int^{2\pi}_0 \mathbf{A}\,d\alpha\,d\beta\,d\gamma$,

where $\mathbf{A}$ is a vector defined as:

$\qquad \left( \begin{array}{c} -\cos(\beta)*\cos(\gamma)*\sin(\alpha)-\cos(\alpha)*\sin(\gamma) \\ \cos(\alpha)*\cos(\beta)*\cos(\gamma)-\sin(\alpha)*\sin(\gamma) \\ \cos(\gamma)*\sin(\beta) \end{array} \right)$

and how when $\mathbf{A}$ becomes a 3x3 matrix (all elements depend only on $\alpha,\beta,\gamma$).

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closed as off-topic by Michael E2, m_goldberg, Artes, MarcoB, Jens Jul 21 '16 at 5:25

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    $\begingroup$ "Wisely" is a bit ambiguous, since it is obvious the integral should be 0 because both Sin and Cos have period 2Pi and every component is integrated over 2Pi. By the way, it seems you have never tried to integrate anything with Mathematica. $\endgroup$ – Artes Jul 20 '16 at 14:57
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    $\begingroup$ Code, at least for the matrix, would make it convenient for those who are interested to test their ideas. Otherwise people who don't have time might just skip it. $\endgroup$ – Michael E2 Jul 20 '16 at 17:19
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Just:

A = {-Cos[b] Cos[g] Sin[a] - Cos[a] Sin[g],Cos[a] Cos[b] Cos[g] - Sin[a] Sin[g], Cos[g] Sin[b]};
Integrate[A, {a, 0, 2 Pi}, {b, 0, Pi}, {g, 0, 2 Pi}]

{0, 0, 0}

If A is a matrix just replace the vector with it.

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