Question:

<p><strong>Step 1:</strong> Analyze the domain [-π/2, π/2] using properties of the floor function [·].</p><p><strong>Step 2:</strong> For x ∈ [-π/2, 0): [x] = -2 (since -π/2 ≈ -1.57, so -2 ≤ x < -1 in this region) and [sin x] = -1 (since -1 ≤ sin x < 0). Thus denominator = -2 + (-1) + 4 = 1.</p><p><strong>Step 3:</strong> For x ∈ [0, π/2]: [x] = 0 (since 0 ≤ x ≤ π/2 ≈ 1.57) and [sin x] = 0 (since 0 ≤ sin x ≤ 1). Thus denominator = 0 + 0 + 4 = 4.</p><p><strong>Step 4:</strong> Split the integral: $$\int_{-\pi/2}^{\pi/2} \frac{dx}{[x] + [\sin x] + 4} = \int_{-\pi/2}^{0} \frac{dx}{1} + \int_{0}^{\pi/2} \frac{dx}{4}$$</p><p><strong>Step 5:</strong> Calculate: $$= [x]_{-\pi/2}^{0} + \frac{1}{4}[x]_{0}^{\pi/2} = (0 - (-\pi/2)) + \frac{1}{4}(\pi/2 - 0) = \frac{\pi}{2} + \frac{\pi}{8} = \frac{5\pi}{8}$$</p><p>∴ Answer: A</p>