Find a function `g:RrarrR`, continuous in `[0,oo)` and positive in `(0,oo)` satisfying `g(1)=1` and `1/2int_0^xg^2(t)dt=1/x(int_0^xg(t)dt)^2`

If g(x) is continuous function in [0, oo) satisfying g(1) = 1. If int_(0)^(x) 2x . g^(2)(t)dt = (int_(0)^(x) 2g(x - t)dt)^(2) , find g(x).

If g(x) is continuous function in [0, oo) satisfying g(1) = 1. If int_(0)^(x) 2x . g^(2)(t)dt = (int_(0)^(x) 2g(x - t)dt)^(2) , find g(x).

Given a function g, continous everywhere such that g (1)=5 and int _(0)^(1) g (t) dt =2. If f (x) =1/2 int _(0) ^(x) (x -t)^(2) g (t) dt, then find the value of f '(1)+f''(1).

Given a funtion g, continous everywhere such that g (1)=5 and int _(0)^(1) g (t) dt =2. If f (x) =1/2 int _(0) ^(x) (x -t)^(2) g (t) dt, then find the value of f ''(1)+f''(1).

Lt_(x to oo) ((int_(0)^(x) e^(t) dt)^(2))/(int_(0)^(x)e^(2t^(2))dt)

Given a function 'g' continous everywhere such that int _(0) ^(1) g (t ) dt =2 and g (1)=5. If f (x ) =1/2 int _(0) ^(x) (x-t) ^(2)g (t)dt, then the vlaue of f'''(1)-f''(1) is:

Given a function 'g' continous everywhere such that int _(0) ^(1) g (t ) dt =2 and g (1)=5. If f (x ) =1/2 int _(0) ^(x) (x-t) ^(2)g (t)dt, then the vlaue of f'(1)-f'(1) is:

Given a function 'g' continous everywhere such that int _(0) ^(1) g (t ) dt =2 and g (1)=5. If f (x ) =1/2 int _(0) ^(x) (x-t) ^(2)g (t)dt, then the vlaue of f''(1)-f'(1) is:

Evaluate int_(0)^(oo)(dt)/(t)