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JEE Main20206 Sep 2020Morning ShiftMathematicsDefinite IntegrationActual

lim x → 1 ∫ 0 ( x - 1 ) 2 t cos t 2 d t ( x - 1 ) sin ( x - 1 )

Options

  1. Ais equal to 1 2 .
  2. Bis equal to 1 .
  3. Cis equal to - 1 2 .
  4. Dis equal to 0 .

Correct answer

D. is equal to 0 .

Step-by-step solution

Applying L'Hospital's Rule, for numerator we have to apply Newton Leibnitz integral rule lim x → 1 2 ( x - 1 ) × ( x - 1 ) 2 cos x - 1 4 - 0 ( x - 1 ) cos ( x - 1 ) + sin ( x - 1 )     0 0 Divide x - 1 = lim x → 1 2 ( x - 1 ) 2 cos ( x - 1 ) 4 cos ( x - 1 ) + sin ( x - 1 ) ( x - 1 ) = 0 1 + 1 = 0

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