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JEE Advanced2011MathematicsDefinite IntegrationActual

The value of _ 2 ^ 3 x x^2 x^2+ ( 6-x^2 ) d x is

Options

  1. A1 4 3 2
  2. B1 2 3 2
  3. C3 2
  4. D1 6 3 2

Correct answer

A. 1 4 3 2

Step-by-step solution

Put x^2=t x d x= d t 2 I= _ 2 ^ 3 t d t 2 t+ ( 6-t) Using, _a^b f(x) d x= _a^b f(a+b-x) d x aligned & I= 1 2 _ 2 ^ 3 ( 2+ 3-t) ( 2+ 3-t)+ d t & I= 1 2 _ 2 ^ 3 ( 6-t) ( 6-t)+ (t) d t & I= _ 2 ^ 3 ( 6-t) ( 6-t)+ t d t ( li ) aligned On adding Eqs. (i) and (ii), we get 2 I= 1 2 _ 2 ^ 3 t+ ( 6-t) ( 6-t)+ t d t gathered 2 I= 1 2 (t)_ 2 ^ 3 = 1 2 ( 3- 2) I= 1 4 ( 3 2 ) gathered

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