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MHT CET202310 May 2023Morning ShiftMathematicsDefinite IntegrationActual

If ₀^ 1 2 x^2 (1-x^2 )^ 3 2 ~d x= k 6 , then the value of k is

Options

  1. A2 3 -
  2. B2 3 +
  3. C3 2 +
  4. D3 2 -

Correct answer

A. 2 3 -

Step-by-step solution

Let I = ₀^ 1 2 x^2 (1-x^2 )^ 3 2 ~d x Put x= aligned & d x= d & (1-x^2 )^ 3 2 = (1- ^2 )^ 3 2 & = ( ^2 )^ 3 2 & = ^3 & I= ₀^ 6 ^2 d ^3 & = ₀^ 6 ^2 d & = ₀^ 6 ( ^2 -1 ) d & =[ ]₀^ 6 -[ ]₀^ 6 & = ( 6 - 0 )- ( 6 -0 ) & = 1 3 - 6 & = 3 3 - 6 & = 2 3 - 6 & But, ₀^ 1 2 x^2 (1-x^2 )^ 3 2 ~d x= k 6 [Given] & k =2 3 - & aligned

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