Definite Integration
Integration involving Greatest Integer and Fractional Part Functions
Grade 12
Question:
<p><strong>256.</strong> \(\displaystyle\int_0^{10} [x]^3 \{x\}\, dx\) is equal to:<br>[Note: Where \([\,]\) and \(\{\,\}\) denotes greatest integer and fractional part functions respectively]</p>
<p>(a) 2025</p>
<p>(b) \(\dfrac{2025}{2}\)</p>
<p>(c) \(\dfrac{2025}{4}\)</p>
<p>(d) \(\dfrac{2025}{8}\)</p>
Step-by-Step Solution
Key Concept: Split the integral at integer points where [x] changes value. For x ∈ [n, n+1), [x] = n (constant) and {x} = x - n, so the integral becomes a sum of integrals where [x] is constant and only {x} varies.
<p><strong>Step 1:</strong> Split the integral using the definition of floor and fractional parts.</p><p>For x ∈ [n, n+1), [x] = n and {x} = x - n.</p><p>∫₀¹⁰ [x]³{x} dx = Σ(n=0 to 9) ∫ₙⁿ⁺¹ n³(x - n) dx</p><p><strong>Step 2:</strong> Evaluate each integral ∫ₙⁿ⁺¹ n³(x - n) dx = n³ ∫₀¹ u du = n³ · [u²/2]₀¹ = n³/2</p><p><strong>Step 3:</strong> Sum over all intervals:</p><p>∫₀¹⁰ [x]³{x} dx = Σ(n=0 to 9) n³/2 = (1/2) · Σ(n=0 to 9) n³</p><p><strong>Step 4:</strong> Use the formula Σ(n=0 to 9) n³ = [9·10/2]² = 45² = 2025</p><p><strong>Step 5:</strong> Therefore, ∫₀¹⁰ [x]³{x} dx = 2025/2 = 1012.5</p><p>∴ Answer: B</p>
Correct Answer: B