If `f(x)` is a function satisfying `f(x+a)+f(x)=0` for all `x in R` and positive constant `a` such that `int_b^(c+b)f(x)dx` is independent of `b ,` then find the least positive value of `c`

Answer

NA

Explanation

`f(x)=int_(0)^(x)sin(t^(2)-t+x)dt`
`=cosx int_(0)^(x)sin(t^(2)-t)dt+sinx int_(0)^(x)cos(t^(2)-t)dt`.
or `f'(x)=-sin int_(0)^(x)sin(t^(2)-t)dt+cosx sin (x^(2)-x)`
`+cosx int_(0)^(x)cos(t^(2)-t)dt+sinx cos (x^(2)-x)`
`=-sinx int_(0)^(x)sin(t^(2)-t)dt+cosx int_(0)^(x)cos(t^(2)-t)dt+sinx^(2)`
or `f''(x)=-sinx sin(x^(2)-x)-cosx int_(0)^(x)sin(t^(2)-t)dt-sinx`
`int_(0)^(x)cos(t^(2)-t)dt+cosxcos(x^(2)-x)+2xsinx^(2)`
`=cosx^(2)-f(x)+2xsinx^(2)`
or `f''(x)+f(x)=cosx^(2)+2xsinx^(2)`