Let `f(x)` be a derivable function satisfying `f(x)=int_0^x e^tsin(x-t)dt and g(x)=f''(x)-f(x)` Then the possible integers in the range of `g(x)` is_______

Answer

3

Explanation

`f(x)int_(0)^(x)e^(t)sin(x-t)dt=int_(0)^(x)e^(x-t)sin(x-(x-t))dt=e^(x)int_(0)^(x)e^(-t)sin t dt`
`:. f'(x)=e^(x)e^(-x)sinx+e^(x)int_(0)^(x)e^(-t)sin t dt`
`=sin x+e^(x)int_(0)^(x)e^(-t)sin t dt`
`:.f''(x)=cosx+e^(x)e^(-x)sinx+e^(x)int_(0)^(x)e^(-t)sin t dt`
`=cosx+sinx+f(x)`
`:.f''(x)-f(x)=cosx+sinx`
Range of `g(x)=f''(x)-f(x)` is `[-sqrt(2),sqrt(2)]`
Number of integers inthe range is 3.