The integral ∫ − 1 / 2 1 / 2 ( [ x ] + ℓ n ( 1 + x 1 − x ) ) d x is equal to ( x is the greatest integer ≤ x )

Question

The integral ∫ − 1 / 2 1 / 2 ( [ x ] + ℓ n ( 1 + x 1 − x ) ) d x is equal to ( x is the greatest integer ≤ x ), − 1 2, 1, 2 ℓ n ( 1 2 ), 0

MARKS App · Accepted Answer

Let, I = ∫ − 1 / 2 1 / 2 ( [ x ] + ℓ n ( 1 + x 1 − x ) ) d x = ∫ − 1 / 2 1 / 2 [ x ] d x + ∫ − 1 / 2 1 / 2 ℓ n ( 1 + x 1 − x ) d x = ∫ − 1 / 2 0 − 1 d x + ∫ 0 1 / 2 0 d x + 0 [ ∵ log ( 1 + x 1 − x ) is a n o d d f u n c t i o n ] = [ − x ] - 1 / 2 0 = 0 − ( 1 2 ) = − 1 2

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