Step-by-Step Solution
Key Concept: The fractional part function (x - [x]) has period 1 and integrates to 1/2 over each unit interval. Partition [0,100] into 100 unit intervals to leverage this periodicity.
<p><strong>Step 1:</strong> Recognize that {x} = x - [x] is the fractional part of x, with period 1.</p><p><strong>Step 2:</strong> For any interval [n, n+1] where n is a non-negative integer:</p><p>∫<sub>n</sub><sup>n+1</sup> (x - [x])dx = ∫<sub>n</sub><sup>n+1</sup> (x - n)dx = [½(x-n)²]<sub>n</sub><sup>n+1</sup> = ½(1)² = ½</p><p><strong>Step 3:</strong> Partition [0, 100] into 100 unit intervals: [0,1], [1,2], ..., [99,100].</p><p><strong>Step 4:</strong> Each interval contributes ½ to the integral:</p><p>∫<sub>0</sub><sup>100</sup> (x - [x])dx = 100 × ½ = <strong>50</strong></p><p><em>Note: The answer is 50, not 45. Please verify the problem statement or expected answer.</em></p>
Correct Answer: 45