Evaluate `-int_(3pi//2)^(pi//2)[2sinx]dx`, when `[.]` denotes the greatest integer function.

Explanation

To evaluate the integral \(-\int_{3\pi/2}^{\pi/2} [2\sin x] \, dx\), where \([.]\) denotes the greatest integer function (GIF), we can follow these steps:

### Step 1: Reverse the Limits of Integration
Using the property of definite integrals, we can reverse the limits of integration when we introduce a negative sign:
\[ -\int_{3\pi/2}^{\pi/2} [2\sin x] \, dx = \int_{\pi/2}^{3\pi/2} [2\sin x] \, dx \]

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