The value of the integral (\int_0^2 \frac{\sin 2x}{[x\pi] + 1} dx) (where ([x]) denotes the greatest integer less than or equal to x) is:

Let I = (\int_0^2 \frac{\sin 2x}{[x\pi] + 1} dx)

Since ([x]) is the greatest integer less than or equal to x, we have:

For 0 ≤ x < 1, [x] = 0
For 1 ≤ x < 2, [x] = 1

Therefore, we can split the integral into two parts:

I = (\int_0^1 \frac{\sin 2x}{[x\pi] + 1} dx + \int_1^2 \frac{\sin 2x}{[x\pi] + 1} dx)

For 0 ≤ x < 1, [xπ] = 0, so the first integral becomes:

(\int_0^1 \frac{\sin 2x}{0 + 1} dx = \int_0^1 \sin 2x dx = \left[ -\frac{\cos 2x}{2} \right]_0^1 = -\frac{\cos 2}{2} + \frac{\cos 0}{2} = \frac{1 - \cos 2}{2})

For 1 ≤ x < 2, [xπ] = 1, so the second integral becomes:

(\int_1^2 \frac{\sin 2x}{1 + 1} dx = \frac{1}{2} \int_1^2 \sin 2x dx = \frac{1}{2} \left[ -\frac{\cos 2x}{2} \right]_1^2 = \frac{1}{4} (-\cos 4 + \cos 2) = \frac{\cos 2 - \cos 4}{4})

Therefore, the value of I is:

I = (\frac{1 - \cos 2}{2} + \frac{\cos 2 - \cos 4}{4} = \frac{2 - 2\cos 2 + \cos 2 - \cos 4}{4} = \frac{2 - \cos 2 - \cos 4}{4})

This expression doesn't directly match any of the given options. However, there might be a calculation error or a different interpretation of the problem statement. It's crucial to double-check the original problem and solution for accuracy. The provided solution steps are complete but the final result is not one of the options presented. Note that the integral is presented differently in the problem statement and solution with different notations. Careful attention should be paid to the way it is written to avoid ambiguity.