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 &#8747; a b x f x + f x + 1 d x is equal to

Aicharya · Accepted Answer

I = 1 a + b &#8747; a b x f x + f x + 1 d x &#160; &#160; &#160; &#160; &#160; &#160; &#8230; 1 x &#8594; a + b - x I = 1 a + b &#8747; a b a + b - x f a + b - x + a + b + 1 - x d x I = 1 a + b &#8747; a b a a + b - x f x + 1 + f x d x &#160; &#160; &#160; &#160; &#160; &#160; &#8230; 2 &#8757; Keeping x &#8594; x + 1 in given relation 1 + 2 2 I = &#8747; a b f x + 1 + f x d x 2 I = &#8747; a b f x + 1 d x + &#8747; a b f x d x &#8747; a b f a + b + 1 - x d x + &#8747; a b f x d x 2 I = 2 &#8747; a b f x d x I = &#8747; a b f x d x Put x = t + 1 &#8658; I = &#8747; a - 1 b - 1 f t + 1 d t = &#8747; a - 1 b - 1 f x + 1 d x

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…

Solution

If f a + b + 1 - x = f x , for all x , where a and b are fixed positive real numbers, then 1 a + b &#8747; a…

Solution

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…