The value of ∫ π 2 π [ 2 sin x ] d x is equal to (where [.] represents the greatest integer function)

Step-by-step solution

Let, I = ∫ π 2 π [ 2 sin x ] d x For π ≤ x ≤ 7 π 6 or 11 π 6 ≤ x ≤ 2 π − 1 ≤ 2 sin x < 0 ⇒ [ 2 sin x ] = − 1 For 7 π 6 ≤ x < 11 π 6 − 2 ≤ 2 sin x < − 1 ⇒ [ 2 sin x ] = − 2 ∴ I = ∫ π 7 π / 6 − 1 d x + ∫ 7 π / 6 11 π / 6 − 2 d x + ∫ 11 π / 6 2 π − 1 d x = ( − 7 π 6 + π ) + 2 ( − 11 π 6 + 7 π 6 ) + ( − 2 π + 11 π 6 ) = − π 6 − 8 π 6 − π 6 = − 10 π 6 = − 5 π 3