If f(a + b – x) = f(x), then ∫ x f(x) dx , x ∈ [a, b] is equal to A. (a + b)/2 ∫ f(b- x)dx, x ∈ [a ,b]

Question

If f(a + b – x) = f(x), then \(\int\limits_{a}^{b} \) x f(x) dx is equal to

A. \(\cfrac{a+b}2\int\limits_{a}^{b} \) f(b- x)dx

B.  \(\cfrac{a+b}2\int\limits_{a}^{b} \) f(b + x)dx

C.  \(\cfrac{b-a}2\int\limits_{a}^{b} \) f(x) dx

D.  \(\cfrac{a+b}2\int\limits_{a}^{b} \) f(x)dx

Answer 1

Correct option is D.  \(\cfrac{a+b}2\int\limits_{a}^{b} \) f(x)dx

Given, f (a + b - x) =f(x)

a + b – x = x

a + b = 2x

x = \(\cfrac{a+b}2\)

Now,