JEE Main 2022 · 29 June · morning · Single correct · +4 / −1

∫₀⁵ cos ( π ( x - [ x/2 ] ) )dx ,
where [t] denotes greatest integer less than or equal to t, is equal to:

  1. (a) -3
  2. (b) -2
  3. (c) 2
  4. (d) 0
Answer and solution

Answer (d) 0

Solution

Maths sir · 4 steps

Break the integral wherever [x/2] jumps, and the integrand turns into plain ±cos π x on each piece.

  1. Find the jumps. [x/2] changes value only at x = 2 and x = 4 inside [0, 5]. So split the integral at those two points.
  2. Simplify the cosine. If [x/2] = n, the integrand is cos(π x - nπ) = (-1)ⁿ cos π x. Each piece is just cos π x with a sign in front.
    Interval[x/2]IntegrandIntegral
    [0, 2)0cos π x(sin 2π - sin 0)/π = 0
    [2, 4)1-cos π x-(sin 4π - sin 2π)/π = 0
    [4, 5]2cos π x(sin 5π - sin 4π)/π = 0
  3. Why each is zero. An antiderivative of cos π x is (sin π x)/π, and sin π x = 0 at every integer. Every limit here is an integer, so every piece gives 0.
  4. So. The total is 0 + 0 + 0 = 0, which is option (d).

The idea to keep. With a greatest integer function inside, split the range at every point where it jumps. On each piece it is a constant, and the integral becomes an ordinary one.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Definite Integration

Asked about 230 times in JEE Main since 2020. This question is from 2022.