If ∫ - a a x + x - 2 d x = 22 , a > 2 and x denotes the greatest integer ≤ x , then ∫ a - a x + x d x is equal to

Step-by-step solution

∫ - a 0 - 2 x + 2 d x + ∫ 0 2 x + 2 - x d x + ∫ 2 a 2 x - 2 d x = 22 ⇒ - x 2 + 2 x | - a 0 + 2 x | 0 2 + x 2 - 2 x | 0 a = 22 ⇒ a 2 + 2 a + 4 + a 2 - 2 a = 22 ⇒ 2 a 2 = 18 ⇒ a = 3   ( a ≠ - 3 ) ∫ 3 - 3 x + x d x = ∫ 3 - 3 x d x + ∫ 3 - 3 x d x = 0 + - - 3 - 2 - 1 + 1 + 2 = 3