NDA 2025 · PYQ · Definite integral / Greatest integer function · hard
Let f(x) = [x²], where [·] is the greatest integer function. What is ∫_{√2}^{2} f(x) dx equal to?
- A.6 − √3 − 2√2✓ Correct
- B.6 − √3 − √2
- C.6 − √3 + 2√2
- D.6 + √3 − 2√2
For x ∈ [√2, √3], x² ∈ [2, 3), so [x²] = 2. For x ∈ [√3, 2], x² ∈ [3, 4), so [x²] = 3. Hence ∫_{√2}^{2} [x²] dx = 2(√3 − √2) + 3(2 − √3) = 2√3 − 2√2 + 6 − 3√3 = 6 − √3 − 2√2.
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