Definite Integration
Integration using greatest integer function
Grade 12
Question:
<p>The value of \(\int_{1}^{a} [x] f'(x) \, dx\), \(a > 1\), where \([x]\) denotes the greatest integer not exceeding \(x\) is</p>
<p>\(af(a) - \{f(1) + f(2) + \cdots + f([a])\}\)</p>
<p>\([a]f(a) - \{f(1) + f(2) + \cdots + f([a])\}\)</p>
<p>\([a]f([a]) - \{f(1) + f(2) + \cdots + f(a)\}\)</p>
<p>\(af([a]) - \{f(1) + f(2) + \cdots + f(a)\}\)</p>
Step-by-Step Solution
Key Concept: Use integration by parts with [x] as the first function and f'(x)dx as the second, recognizing that [x] is constant on intervals [n, n+1). Split the integral at integer points where [x] jumps.
<p><strong>Step 1:</strong> Recognize that [x] is a step function. For any x ∈ [n, n+1), [x] = n where n is an integer.</p><p><strong>Step 2:</strong> Apply integration by parts: Let u = [x], dv = f'(x)dx. Then du = 0 (almost everywhere) and v = f(x).</p><p><strong>Step 3:</strong> Using integration by parts formula ∫u dv = uv - ∫v du, and noting the boundary behavior:</p><p>∫₁ᵃ [x]f'(x)dx = [x]·f(x)|₁ᵃ - ∫₁ᵃ f(x)·0 dx</p><p><strong>Step 4:</strong> Evaluate the boundary term. At x = a: [a]·f(a). At x = 1: [1]·f(1) = 1·f(1) = f(1).</p><p>∫₁ᵃ [x]f'(x)dx = [a]f(a) - f(1)</p><p>∴ Answer: B</p>
Correct Answer: B