If f a + b + 1 - x = f x , for all x , where a and b are fixed positive real numbers, then 1 a + b ∫ a b x f x + f x + 1 d x is equal to

Question

If f a + b + 1 - x = f x , for all x , where a and b are fixed positive real numbers, then 1 a + b ∫ a b x f x + f x + 1 d x is equal to, ∫ a - 1 b - 1 f x + 1 d x, ∫ a - 1 b - 1 f x d x, ∫ a + 1 b + 1 f x d x, ∫ a + 1 b + 1 f x + 1 d x

MARKS App · Accepted Answer

I = 1 a + b ∫ a b x f x + f x + 1 d x … 1 x → a + b - x I = 1 a + b ∫ a b a + b - x f a + b - x + a + b + 1 - x d x I = 1 a + b ∫ a b a a + b - x f x + 1 + f x d x … 2 ∵ Keeping x → x + 1 in given relation 1 + 2 2 I = ∫ a b f x + 1 + f x d x 2 I = ∫ a b f x + 1 d x + ∫ a b f x d x ∫ a b f a + b + 1 - x d x + ∫ a b f x d x 2 I = 2 ∫ a b f x d x I = ∫ a b f x d x Put x = t + 1 ⇒ I = ∫ a - 1 b - 1 f t + 1 d t = ∫ a - 1 b - 1 f x + 1 d x

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