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JEE Main20198 Apr 2019Morning ShiftMathematicsDefinite IntegrationActual

If f x = 2 - x c o s x 2 + x c o s x and g ( x ) = log e ⁡ x , then the value of the integral ∫ - π 4 π 4 g f x d x is

Options

  1. Alog e ⁡ e
  2. Blog e ⁡ 2
  3. Clog e ⁡ 1
  4. Dlog e ⁡ 3

Correct answer

C. log e ⁡ 1

Step-by-step solution

Given, f x = 2 - x c o s x 2 + x c o s x , g ( x ) = log e ⁡ x ⇒ g f x = l o g e 2 - x c o s x 2 + x c o s x ⇒ g f - x = ⁡ l o g e 2 - ( - x ) c o s ⁡ ( - x ) 2 + ( - x ) c o s ⁡ ( - x ) ⇒ g f - x = log e 2 + x cos x 2 - x cos x ⇒ g f - x = - log 2 - x cos x 2 + x cos x ⇒ g f - x = – g ( f x ) Hence, g ( f x ) is an odd function. By using the property of definite integration, ∫ - a a f x d x = 2 ∫ 0 a f x d x ,     f - x = f x 0 , f -

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