<p>If \([\;]\) denotes the greatest integer function, then the integral \(\displaystyle\int_0^{\pi}[\cos x]\,dx\) is equal to</p>
Step-by-Step Solution
Key Concept: The greatest integer function [cos x] takes discrete values based on the range of cos x on [0,π]. Since cos x decreases from 1 to -1, we must identify intervals where [cos x] is constant, then integrate piecewise.
<p><strong>Step 1:</strong> Determine the range of cos x on [0,π].<br>As x goes from 0 to π, cos x decreases from cos(0) = 1 to cos(π) = -1.</p><p><strong>Step 2:</strong> Identify intervals where [cos x] is constant.<br>• At x = 0: cos(0) = 1, so [cos x] = [1] = 1 (single point)<br>• For x ∈ (0, π/2): 0 < cos x < 1, so [cos x] = 0<br>• At x = π/2: cos(π/2) = 0, so [cos x] = [0] = 0<br>• For x ∈ (π/2, π): -1 < cos x < 0, so [cos x] = -1<br>• At x = π: cos(π) = -1, so [cos x] = [-1] = -1 (single point)</p><p><strong>Step 3:</strong> Set up the piecewise integral.<br>$$\int_0^{\pi}[\cos x]\,dx = \int_0^{\pi/2} 0\,dx + \int_{\pi/2}^{\pi} (-1)\,dx$$</p><p><strong>Step 4:</strong> Evaluate.<br>$$= 0 + (-1)\left(\pi - \frac{\pi}{2}\right) = -\frac{\pi}{2}$$</p><p>∴ Answer: C (which equals $-\frac{\pi}{2}$)</p>
Correct Answer: C