Question:

**Step 1: Analyze the behavior of $[\cos x]$ over the interval $[0, 2\pi]$.** The greatest integer function $[y]$ returns the largest integer less than or equal to $y$. We identify the intervals where $[\cos x]$ takes a constant integer value. * For $x \in [0, \pi/2)$: $\cos x \in (0, 1]$. Thus, $[\cos x] = 0$. * For $x \in (\pi/2, 3\pi/2)$: $\cos x \in [-1, 0)$. Thus, $[\cos x] = -1$. * For $x \in (3\pi/2, 2\pi]$: $\cos x \in (0, 1]$. Thus, $[\cos x] = 0$. The values of $[\cos x]$ at the points $x=0, \pi/2, \pi, 3\pi/2, 2\pi$ are $1, 0, -1, 0, 1$ respectively. Since these are single points, they are of measure zero and do not affect the value of the definite integral. **Step 2: Evaluate the definite integral.** The integral can be split based on the constant values of $[\cos x]$ over the identified intervals: $$ \int_0^{2\pi} [\cos x] \, dx = \int_0^{\pi/2} [\cos x] \, dx + \int_{\pi/2}^{3\pi/2} [\cos x] \, dx + \int_{3\pi/2}^{2\pi} [\cos x] \, dx $$ Substitute the constant values of $[\cos x]$ for each interval: $$ \int_0^{2\pi} [\cos x] \, dx = \int_0^{\pi/2} 0 \, dx + \int_{\pi/2}^{3\pi/2} (-1) \, dx + \int_{3\pi/2}^{2\pi} 0 \, dx $$ $$ \int_0^{2\pi} [\cos x] \, dx = 0 + (-1) \left[ x \right]_{\pi/2}^{3\pi/2} + 0 $$ $$ \int_0^{2\pi} [\cos x] \, dx = (-1) \left( \frac{3\pi}{2} - \frac{\pi}{2} \right) $$ $$ \int_0^{2\pi} [\cos x] \, dx = (-1) (\pi) $$ $$ \int_0^{2\pi} [\cos x] \, dx = -\pi $$