Let `f(x)=int_0^x e^t.sin(x-t) dt and g(x)=f(x)+f''(x).` Which of the following statements are correct ?

Let f(x) be a derivable function satisfying f(x)=int_(0)^(x)e^(t)sin(x-t)dt and g(x)=f'(x)-f(x) Then the possible integers in the range of g(x) is

Let f(x)=int_(0)^(1)|x-t|dt, then

If f(x) = int_(0)^(x)t sin t dt , then f'(x) is

If f(x)=int_0^x (sint)/(t)dt,xgt0, then

Let psi_1:[0,oo] to R , psi_2:[0,oo) to R , f:[0,oo) to R and g :[0,oo) to R be functions such that f(0)=g(0)=0 psi_1(x)=e^-x+x , x ge 0 , psi_2(x)=x^2-2x-2e^-x+2 , x ge0 f(x)=int_(-x)^x (abs(t)-t^2)e^(-t^2)dt , x gt 0 g(x)=int_0^(x^2) (sqrtt)e^-t dt , x gt o Which of the following statements is TRUE

Let psi_1:[0,oo] to R , psi_2:[0,oo) to R , f:[0,oo) to R and g :[0,oo) to R be functions such that f(0)=g(0)=0 psi_1(x)=e^-x+x , x ge 0 , psi_2(x)=x^2-2x-2e^-x+2 , x ge0 f(x)=int_(-x)^x (abs(t)-t^2)e^(-t^2)dt , x gt 0 g(x)=int_0^(x^2) (sqrtt)e^-t dt , x gt o Which of the following statements is TRUE

Let f be a continuous function on R and satisfies f(x)=e^(x)+int_(0)^(1)e^(x)f(t)dt, then which of the following is(are) correct?