Let `f(x)` is a derivable function satisfying `xf(x) - int_0 ^xf(t)dt = x + ln ( sqrt(x^2 +1) - x) ` with `f(0) = ln2` Let ` g(x) = x f'(x)`, Then

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 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 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 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 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_______

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) 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_______

if f(x) is a differential function such that f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e , then Q. int_(0)^(1)f(x)dx=

if f(x) is a differential function such that f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e , then Q. int_(0)^(1)f(x)dx=

Suppose that f(x) is a differentiable function such that f'(x) is continuous, f'(0) = 1 and f"(0) does not exist. Let g(x) = xf'(x). Then