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AP EAMCET202120 Aug 2021Evening ShiftMathematicsIndefinite IntegrationActual

If the primitive of cos ( log x ) is f ( x ) cos ( g ( x ) ) + sin ( h ( x ) ) , then which among the following is true?

Options

  1. Ah ' ( x ) = - 1 x
  2. Bf ' ( x ) = 1 2
  3. Cg ' ( x ) = log ( x )
  4. Dh ( x ) = x 2

Correct answer

B. f ' ( x ) = 1 2

Step-by-step solution

Primitive means integration ⇒ ∫ cos log x d x = f x cos g x + sin h x ⇒ I = ∫ cos log x d x By parts formula ∫ u · v   d x = u ∫ v   d x - ∫ d d x u ∫ v d x d x ⇒ I = ∫ 1 . cos log x d x ⇒ I = cos log x · x - ∫ - sin log x x x   d x ⇒ I = x cos log x + ∫ sin log x d x ⇒ I = x cos log x + sin log x x - ∫ cos log x x x   d x ⇒ I = x cos log x + x sin log x - ∫ cos log x d x ⇒

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