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JEE Main202120 Jul 2021Evening ShiftMathematicsDefinite IntegrationActual

If [ x ] denotes the greatest integer less than or equal to x , then the value of the integral ∫ - π / 2 π / 2 x - sin x d x is equal to:

Options

  1. A- π
  2. Bπ
  3. C0
  4. D1

Correct answer

A. - π

Step-by-step solution

We have, I = ∫ - π / 2 π / 2 x - sin x d x   … i ⇒ I = ∫ - π 2 - 1 x - sin x d x + ∫ - 1 0 x - sin x d x + ∫ 0 1 x - sin x d x + ∫ 1 π 2 x - sin x d x ⇒ I = ∫ - π / 2 - 1 - 2 - sin x d x + ∫ - 1 0 - 1 - sin x d x + ∫ 0 1 0 - sin x d x + ∫ 1 π 2 1 - sin x d x ⇒ I = - 2 ∫ - π / 2 - 1 d x - ∫ - 1 0 d x - 1 ∫ 0 1 d x + ∫ 1 π 2 0 · d x ⇒ I = - 2 - 1 + π 2 - 1

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