The value of $\mathop \smallint \limits_{ - 1}^1 \,\,{\sin ^{ - 1}}\,\,\left[ {{x^2}\,\, + \,\,\frac{1}{2}} \right]\,\,dx\,\, + \,\,\mathop \smallint \limits_{ - 1}^1 \,\,{\cos ^{ - 1}}\,\,\left[ {{x^2} - \,\,\frac{1}{2}} \right]\,\,dx$, where $\left[ {\, \cdot \,} \right]$ denotes the greatest integer function, is

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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The value of

\int \limits_{ - 1}^1 \,\,{\sin ^{ - 1}}\,\,\left[ {{x^2}\,\, + \,\,\frac{1}{2}} \right]\,\,dx\,\, + \,\,\int \limits_{ - 1}^1 \,\,{\cos ^{ - 1}}\,\,\left[ {{x^2} - \,\,\frac{1}{2}} \right]\,\,dx

, where

\left[ {\, \cdot \,} \right]

denotes the greatest integer function, is

View answer and explanation

Correct option: B

2\,\pi

Option B (2π) is correct because the sum of the two integrals simplifies to 2π after evaluating the greatest integer functions and using trigonometric identities.

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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