Let f(x) be a function defined on R satisfyin `f(x) =f(1-x)` for all ` x in R`. Then `int_(-1//2)^(1//2) f(x+(1)/(2))sin x dx` equals

Options

A. `underset(-1//2)overset(1//2)int f(x+(1)/(2))sin x dx`
B. `2underset(-1//2)overset(1//2)int f(x+(1)/(2))sin x dx`
C. `f(x) sind x dx`
D. none of these

Answer

D

Explanation

Log `g(x)=I(x+(1)/(2)) sin x`, Then,
`g(-x)=-f(-x+(1)/(2))sin (-x)`
`g(-x)=-f(1+x+(1)/(2))sinx ["Using "f(x) =f(1-x)]`
`g(-x)=-f(x+(1)/(2))sin x-g(x)`
`underset(-1//2)overset(1//2) itn f(x+(1)/(2)) sin x dx=0`