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

Explanation

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

### Step 1: Change the limits of integration
We can change the limits of integration by switching the upper and lower limits, which introduces a negative sign. Thus, we have:
\[ -\int_{\frac{3\pi}{2}}^{\frac{\pi}{2}} [2\sin x] \, dx = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} [2\sin x] \, dx \]

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