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 using the property that [x] is constant on each unit interval [n, n+1), and {x} = x - [x]. On [n, n+1), we have [x] = n and {x} = x - n.
<p><strong>Step 1:</strong> Split the integral into unit intervals where [x] is constant:</p><p>∫₀¹⁰ [x]³{x} dx = Σₙ₌₀⁹ ∫ₙⁿ⁺¹ n³(x-n) dx</p><p><strong>Step 2:</strong> On interval [n, n+1), [x] = n and {x} = x - n. Evaluate the generic integral:</p><p>∫ₙⁿ⁺¹ n³(x-n) dx = n³ ∫₀¹ u du (substituting u = x - n)</p><p>= n³ · [u²/2]₀¹ = n³/2</p><p><strong>Step 3:</strong> Sum over all 10 intervals:</p><p>Σₙ₌₀⁹ n³/2 = (1/2) Σₙ₌₀⁹ n³</p><p><strong>Step 4:</strong> Use the formula Σₙ₌₀⁹ n³ = [9(10)/2]² = 45² = 2025</p><p>∴ Answer = (1/2) × 2025 = <strong>2025/2 or 1012.5</strong></p>
Correct Answer: C