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NDA 2 2025 Maths, question 92

Topic: Integral calculus

Q92

What is ∫nn+1(x−[x])dx\int _{n}^{n+1}(x-[x]) dx, where [⋅][\cdot ] is the greatest integer function and nn is natural number?

  1. (a) 4n+12\frac{4n+1}{2}
  2. (b) 2n+12\frac{2n+1}{2}
  3. (c) 12\frac{1}{2}Correct answer
  4. (d) 11

Solution.

Answer: (c) 12\frac{1}{2}

  1. For n≤x<n+1, [x]=n.
  2. The integral is ∫ from n to n+1 of (x−n)dx=1/2.
  3. The endpoint value at n+1 does not affect an integral; geometrically this is a triangle of base and height 1.