Let f be a continuous function satisfying ∫ 0 t 2 f ( x ) + x 2 dx = 4 3 t 3 , ∀ t > 0 . Then f π 2 4 is equal to

Step-by-step solution

Given equation is ∫ 0 t 2 f ( x ) + x 2 dx = 4 3 t 3 , ∀ t > 0 According to Newton Leibnitz theorem we have d d x ∫ u x v x f t d t = f v x × v ' x - f u x × u ' x Apply Newtons Leibnitz theorem in the given equation. ⇒ f t 2 + t 4 2 t - 0 = 4 t 2 ⇒ f t 2 + t 4 = 2 t ⇒ f x 2 = - x 4 + 2 x ⇒ f ( x ) = - x 2 + 2 x ⇒ f π 2 4 = - π 4 4 2 + 2 × π 2 = - π 4 16 + π = π 1 - π 3 16 Hence this is the correct option.