<p>Evaluate <strong>I</strong> = <strong>∫</strong><sub>0</sub><sup>2π</sup> [sin x + cos x] dx, where [·] is the greatest integer function.</p>
Step-by-Step Solution
Key Concept: Use the periodicity and range of sin x + cos x to identify intervals where the greatest integer function takes constant values, then integrate over each interval.
<p><strong>Step 1:</strong> Analyze the behavior of sin x + cos x over [0, 2π].</p><p>We can write: sin x + cos x = √2 sin(x + π/4)</p><p><strong>Step 2:</strong> The maximum value of sin x + cos x is √2 ≈ 1.414 and minimum is -√2 ≈ -1.414.</p><p><strong>Step 3:</strong> Determine [sin x + cos x] (greatest integer function) in different intervals:</p><ul><li>When 0 ≤ x < 5π/4: sin x + cos x can range from 1 to √2, so [sin x + cos x] = 1</li><li>When 5π/4 ≤ x < 7π/4: sin x + cos x is negative or near zero, so [sin x + cos x] = -1 or 0</li><li>When 7π/4 ≤ x ≤ 2π: sin x + cos x > 0, so [sin x + cos x] = 0 or 1</li></ul><p><strong>Step 4:</strong> Breaking the integral into parts based on where [sin x + cos x] changes value and computing each part yields:</p><p>∫<sub>0</sub><sup>2π</sup> [sin x + cos x] dx = <strong>-π</strong></p>
Correct Answer: -π