The value of \( \int_{0}^{\pi}[2 \sin x] d x \), where [.] denotes the greatest integer function...

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The value of \( \int_{0}^{\pi}[2 \sin x] d x \), where [.] denotes the greatest integer function, is
(A) \( \frac{\pi}{3} \)
(B) \( \frac{2 \pi}{3} \)
(C) \( \frac{\pi}{6} \)
(D) None of these
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