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NTA Abhyas JEE Main2020MathematicsDefinite IntegrationPractice

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

Options

  1. A− 1 2
  2. B1
  3. C2 ℓ n ( 1 2 )
  4. D0

Correct answer

A. − 1 2

Step-by-step solution

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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