Let f x and g x be two functions satisfying f x 2 + g ( 4 - x ) = 4 x 3 and g 4 - x + g x = 0 , then the value of ∫ - 4 4 f x 2 d x is

Step-by-step solution

I = ∫ - 4 4 f x 2 d x = 2 ∫ 0 4 f x 2 d x Even function = 2 ∫ 0 4 4 x 3 - g 4 - x d x = 2 4 x 4 4 | 0 4 - ∫ 0 4 g 4 - x d x = 512 - 2 I 1 I 1 = ∫ 0 4 g 4 - x d x = ∫ 0 2 g 4 - x d x + ∫ 2 4 g 4 - x d x Let I 2 = ∫ 2 4 g 4 - x d x If 4 - x = t then I 2 = - ∫ 2 0 g t d t = ∫ 0 2 g t d t = ∫ 0 2 g x d x So, I 1 = ∫ 0 2 g 4 - x d x + ∫ 0 2 g x d x = 0 Hence, I = 512