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JEE Main2012MathematicsDefinite IntegrationActual

If d d x G(x)= e^ x x , x (0, / 2) , then _ 1 / 4 ^ 1 / 2 2 x e^ ( x^2 ) d x is equal to

Options

  1. AG( / 4)-G( / 16)
  2. B2[G( / 4)-G( / 16)]
  3. C[G(1 / 2)-G(1 / 4)]
  4. DG(1 / 2 )-G(1 / 2)

Correct answer

A. G( / 4)-G( / 16)

Step-by-step solution

Let d d x G(x)= e^ x x , x (0, 2 ) Now, aligned I & = _ 1 / 4 ^ 1 / 2 2 x e^ x^2 d x & = _ 1 / 4 ^ 1 / 2 2 x x^2 e^ x^2 d x aligned Let x^2=t 2 x d x=d t When x= 1 2 , t= 4 and x= 1 4 , t= 16 aligned I & = _ / 16 ^ / 4 e^ t t d t=g(t) array l 4 16 array & =G ( 4 )-G ( 16 ) aligned

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