Let f : R → R be a continuous function such that f x + f x + 1 = 2 for all x ∈ R . If I 1 = ∫ 0 8 f x d x and I 2 = ∫ - 1 3 f x d x , then the value of I 1 + 2 I 2 is equal to ________.

Step-by-step solution

f x + f x + 1 = 2 Replace x   by   x + 1 f x + 1 + f x + 2 = 2 By subtracting the equations, we get f x = f x + 2 ⇒ f x is periodic with period = 2 I 1 = ∫ 0 8 f x d x = 4 ∫ 0 2 f x d x = 4 ∫ 0 1 f x + f 1 + x d x = 8 Similarly, I 2 = 2 × 2 = 4 Hence, I 1 + 2 I 2 = 16