Definite Integration
Greatest Integer Function Integration
Grade 12
Question:
<p>The value of <span>\(\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{dx}{[x]+[\sin x]+4}\)</span>, where <span>\([t]\)</span> denotes the greatest integer less than or equal to <span>\(t\)</span>, is:</p>
<p>\(\dfrac{1}{12}(7\pi+5)\)</p>
<p>\(\dfrac{1}{12}(7\pi-5)\)</p>
<p>\(\dfrac{3}{20}(4\pi-3)\)</p>
<p>\(\dfrac{3}{10}(4\pi-3)\)</p>
Step-by-Step Solution
Key Concept: Split the integral using symmetry properties of the floor function. For x ∈ [-π/2, π/2], [x] and [sin x] have specific constant values in different subintervals, and the integrand exhibits symmetry that simplifies computation.
<p><strong>Step 1:</strong> Analyze the domain [-π/2, π/2]. For this interval:</p><ul><li>When x ∈ [-π/2, 0): [x] = -1 and [sin x] = -1 (since -1 ≤ sin x < 0)</li><li>When x ∈ [0, π/2]: [x] = 0 and [sin x] = 0 (since 0 ≤ sin x ≤ 1, but sin x < 1 except at endpoints)</li></ul><p><strong>Step 2:</strong> For x ∈ [-π/2, 0): denominator = -1 + (-1) + 4 = 2</p><p>For x ∈ [0, π/2]: denominator = 0 + 0 + 4 = 4</p><p><strong>Step 3:</strong> Split the integral:</p><p>∫<sub>-π/2</sub><sup>π/2</sup> dx/([x]+[sin x]+4) = ∫<sub>-π/2</sub><sup>0</sup> dx/2 + ∫<sub>0</sub><sup>π/2</sup> dx/4</p><p><strong>Step 4:</strong> Evaluate each part:</p><p>= [x/2]<sub>-π/2</sub><sup>0</sup> + [x/4]<sub>0</sub><sup>π/2</sup></p><p>= (0 - (-π/4)) + (π/8 - 0)</p><p>= π/4 + π/8 = 3π/8</p><p>∴ Answer: D (3π/8)</p>
Correct Answer: D