The value of ∫ - 1 1 x - x d x (where [.] denotes greatest integer function) is

Step-by-step solution

∫ − 1 1 ( x − [ x ] ) d x = ∫ − 1 1 x d x − ∫ − 1 1 [ x ] d x = [ x 2 2 ] − 1 1 − [ ∫ − 1 0 [ x ] d x + ∫ 0 1 [ x ] d x ] = 1 2 [ 1 − 1 ] − [ ∫ − 1 0 ( − 1 ) d x + ∫ 0 1 0. d x ] [ If − 1 ≤ x < 0 , [ x ] = − 1 If 0 ≤ x < 1 , [ x ] = 0 ] = 0 − [ − x ] − 1 0 − 0 = 0 − [ − 0 − ( 1 ) ] = 1