If `underset(sinx)overset(1)intt^(2) (f(t)) dt = (1-sinx)`, then `f ((1)/(sqrt(3)))` is

If int_(sinx)^(1)t^(2) (f(t)) dt = (1-sinx) , then f ((1)/(sqrt(3))) is

If y = underset(u(x))overset(v(x))intf(t) dt , let us define (dy)/(dx) in a different manner as (dy)/(dx) = v'(x) f^(2)(v(x)) - u'(x) f^(2)(u(x)) alnd the equation of the tangent at (a,b) as y -b = (dy/dx)_((a,b)) (x-a) If F(x) = underset(1)overset(x)inte^(t^(2)//2)(1-t^(2))dt , then d/(dx) F(x) at x = 1 is

Let f:(0,1) in (0,1) be a differenttiable function such that f(x)ne 0 for all x in (0,1) and f((1)/(2))=(sqrt(3))/(2) . Suppose for all x, underset(x to x)lim(overset(1)underset(0)int sqrt(1(f(s))^(2))dxoverset(x)underset(0)int sqrt(1(f(s))^(2))ds)/(f(t)-f(x))=f(x) Then, the value of f((1)/(4)) belongs to

If overset(4)underset(-1)int f(x)=4 and overset(4)underset(2)int (3-f(x))dx=7 , then the value of overset(-1)underset(2)int f(x)dx , is

Let F(x)=int_(sinx)^(cosx)e^((1+sin^(-1)(t))dt on [0,(pi)/(2)] , then

The value of overset(sin^(2)x)underset(0)int sin^(-1)sqrt(t)dt+overset(cos^(2)x)underset(0)int cos^(-1)sqrt(t)dt , is

If f is continuous function and F(x)=int_0^x((2t+3). int_t^2 f(u)du)\ dt , then |(F^(2))/(f(2))| is equal to____ .

If f(x) = [x] [sinx] in (-1,1) then f(x) is

If int_(0)^(x)f(t)dt = x^(2)-int_(0)^(x^(2))(f(t))/(t)dt then find f(1) .

If F(x) =(1)/(x^(2))overset(x)underset(4)int [4t^(2)-2F'(t)]dt then F'(4) equals