<p>The value of \(\int_0^{\pi} [\cos x]\,dx\) (where \([\cdot]\) denotes the greatest integer function) is:</p>
Step-by-Step Solution
Key Concept: The greatest integer function [cos x] takes constant integer values on different intervals of [0, π]. Identify where cos x crosses integer boundaries (specifically where cos x = 0 and cos x = -1) to partition the integral into regions where [cos x] is constant.
<p><strong>Step 1:</strong> Analyze the behavior of cos x on [0, π].</p><p>• For x ∈ [0, π/2): cos x ∈ (0, 1], so [cos x] = 0</p><p>• At x = π/2: cos x = 0, so [cos x] = 0</p><p>• For x ∈ (π/2, π]: cos x ∈ [-1, 0), so [cos x] = -1</p><p><strong>Step 2:</strong> Split the integral based on where [cos x] changes.</p><p>$$\int_0^{\pi} [\cos x]\,dx = \int_0^{\pi/2} 0\,dx + \int_{\pi/2}^{\pi} (-1)\,dx$$</p><p><strong>Step 3:</strong> Evaluate each part.</p><p>$$= 0 + (-1)\left[x\right]_{\pi/2}^{\pi} = -1\left(\pi - \frac{\pi}{2}\right) = -\frac{\pi}{2}$$</p><p>∴ Answer: <strong>-π/2</strong> (Option A)</p>
Correct Answer: A