Let f : R → R be a function such that f 2 - x = f 2 + x and f 4 - x = f 4 + x , for all x ∈ R and ∫ 0 2 f x d x = 5 . Then the value of ∫ 10 50 f x d x is

Step-by-step solution

f 2 - x = f 2 + x and f 4 - x = f ( 4 + x ) Then curve is symmetric about x = 2 and x = 4 ∴       The curve is periodic with period 4 Given ∫ 0 2 f x d x = 5 ∫ 0 4 f x d x = 2 ∫ 0 2 f x d x = 10 ∫ 10 50 f x d x = ∫ 2 4 + 2 12 4 + 2 f x d x = 12 - 2 ∫ 0 4 f x d x = 10 × 10 = 100