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`

int_(0)^(1)t^(5)*sqrt(1-t^(2))*dt

2int_(0)^(t)(1-cos t)/(t)dt

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).

The function f:[0,1]rarr R is continuous on [0,1] and int_(0)^(x)f(t)dt=int_(x)^(1)f(t)dt . Prove that f(x)=0 for all x in[0,1]

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)

A continuous function f(x) satisfies the relation f(x)=e^(x)+int_(0)^(1)e^(x)f(t)dt then f(1)=

Let f be a non-negative function in [0, 1] and twice differentiate in (0, 1). If int_0^x sqrt(1-(f'(t))^(2))dt=int_0^x f(t)dt , 0 lexle1 and f(0)=0 then the value of lim_(x to0)int_0^xf(t)/x^2 dt is