Verified Solution • 2025–26 Session moderate

QJEEM21Mac797: JEE Main 2021 Mathematics - Class 12 JEEM21Mac797 Solution: If [x] denotes the greatest integer less than or equal to x,…

If [x] denotes the greatest integer less than or equal to x, then the value of the integral ∫−π/2π/2[[x]−sin⁡x]dx\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx} is equal to :

Multiple Choice Options

  • A −- π\pi
  • B π\pi
  • C 0
  • D 1

Step-by-Step Verified Solution

I=∫−π/2π/2[[x]−sin⁡x]dxI = \int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx}

=∫−π/2π/2([x]+[−sin⁡x])dx= \int_{ - \pi /2}^{\pi /2} {\left( {[x] + [ - \sin x]} \right)dx}

=∫0π/2([x]+[−sin⁡x]+[−x]+[sin⁡x])dx= \int_0^{\pi /2} {\left( {[x] + [ - \sin x] + [ - x] + [\sin x]} \right)} dx

=∫0π/2(−2)dx= \int_0^{\pi /2} {( - 2)dx}

=−π= - \pi

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