If x is the greatest integer ≤ x , then π 2 ∫ 0 2 sin π x 2 x - x [ x ] d x is equal to :

Step-by-step solution

Let, I = π 2 ∫ 0 2 sin π x 2 x - x [ x ] d x I = π 2 ∫ 0 1 sin π x 2 x - x [ x ] d x + π 2 ∫ 1 2 sin π x 2 x - x [ x ] d x I = π 2 ∫ 0 1 sin π x 2 x 0 d x + π 2 ∫ 1 2 sin π x 2 x - 1 1 d x I = π 2 ∫ 0 1 sin π x 2 d x + π 2 ∫ 1 2 sin π x 2 x - 1 d x I = π 2 ∫ 0 1 sin π x 2 d x + π 2 x ∫ 1 2 sin π x 2 d x - ∫ 1 2 d d x x ∫ 1 2 sin π x 2 d x d x - π 2 &#8