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JEE Main2013MathematicsIndefinite IntegrationActual

If ∫ f ( x ) d x = ψ ( x ) , then ∫ x 5 f ( x 3 ) d x , is equal to

Options

  1. A1 3 x 3 ψ ( x 3 ) − ∫ x 2 ψ ( x 3 ) d x + c
  2. B1 3 [ x 3 ψ ( x 3 ) − ∫ x 3 ψ ( x 3 ) d x ] + c
  3. C1 3 [ x 3 ψ ( x 3 ) − ∫ x 2 ψ ( x 3 ) d x ] + c
  4. D1 3 x 3 ψ ( x 3 ) − 3 ∫ x 3 ψ ( x 3 ) d x + c

Correct answer

A. 1 3 x 3 ψ ( x 3 ) − ∫ x 2 ψ ( x 3 ) d x + c

Step-by-step solution

∫ x 5 f x 3 d x = ∫ x 3 . x 2 f x 3 d x   Applying integration by parts, we get = x 3 ∫ x 2 f x 3 d x - ∫ d d x x 3 ∫ x 2 f x 3 d x = 1 3 x 3 ψ x 3 - ∫ x 2 ψ x 3 d x + c , where c is the constant of integration.

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