Let `f:(0,oo)vec(0,oo)` be a differentiable function satisfying, `xint_0^x(1-t)f(t)dt=int_0^x tf(t)dtx in R^+a n df(1)=1.` Determine `f(x)dot`

Let f(x) be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt then int_(0)^(1)f(x)dx=

Let f(x) be a differentiable function satisfying f(x)=int_(0)^(x)e^((2tx-t^(2)))cos(x-t)dt , then find the value of f''(0) .

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of lim_(x to 0)(cosx)/(f(x)) is equal to where [.] denotes greatest integer function

If int_(0)^(1) f(t)dt=x^2+int_(0)^(1) t^2f(t)dt , then f'(1/2)is

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of int_(0)^(pi//2) f(x)dx lies in the interval

If f(x)=int_(0)^(x)tf(t)dt+2, then

If int_(0)^(x) f(t)dt=x^(2)+2x-int_(0)^(x) tf(t)dt, x in (0,oo) . Then, f(x) is