If \( f(x)=\left|\begin{array}{ccc}\sin 2 x(1+2 \cos x) & \sin 2 x ...

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If \( f(x)=\left|\begin{array}{ccc}\sin 2 x(1+2 \cos x) & \sin 2 x & \sin 3 x \\ 3+4 \sin x & 3 & 4 \sin x \\ 1+\sin x & \sin x & 1\end{array}\right| \)
then the value of \( \int_{0}^{\pi / 2} f(x) d x \) is
(a) 3
(b) \( 2 / 3 \)
(c) \( 1 / 3 \)
(d) 0
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