Let 5 f x + 4 f 1 x = 1 x + 3 , x > 0 . Then 18 ∫ 1 2 f x d x is equal to

Step-by-step solution

Given: 5 f x + 4 f 1 x = 1 x + 3       . . . . 1 Replace x → 1 x 5 f 1 x + 4 f x = x + 3       . . . . 2 Eliminating f 1 x from 1   &   2 , we get 9 f x = 5 x + 15 - 4 x - 12 ⇒ 9 f x = 5 x - 4 x + 3 ⇒ 18 f x = 10 x - 8 x + 6 ⇒ 18 ∫ 1 2 f x d x = ∫ 1 2 10 x - 8 x + 6 d x ⇒ 18 ∫ 1 2 f x d x = 10 log x - 4 x 2 + 6 x 1 2 ⇒ 18 ∫ 1 2 f x d x = 10 log 2 - 4 - 10 log 1 + 2 ⇒ 18 ∫ 1 2 f x d x = 10 log e 2 - 6