For any real number x , let x denote the largest integer less than or equal to x . Let f be a real-valued function defined on the interval - 10 , 10 by f x = x - x , if x is odd 1 + x - x , if x is even Then, the value of π 2 10 ∫ - 10 10 f x cosπ x d x is

Step-by-step solution

f x = x - 1 , 1 - x , 1 ≤ x < 2 0 ≤ x < 1 f x is periodic with period 2 , and is even. ∴ I = ∫ - 10 10   f x   cosπ x   d x =   2 ∫ 0 10 f x   cosπ x   d x = 2 × 5 ∫ 0 2 f x   cosπ x   d x = 10   ∫ 0 1 1 - x   cosπ x   d x + ∫ 1 2 x - 1   cosπ x   d x = 10 I 1 + I 2 I 2 = ∫ 1 2 x - 1   cosπ x   d x Put x - 1 = t I 2 = - ∫ 0 1 t cosπ t