Let `g(x)` be differentiable on `R` and `int_(sint)^1x^2g(x)dx=(1-sint),` where `t in (0,pi/2)dot` Then the value of `g(1/(sqrt(2)))` is____

If int_(0)^(x)f(x)sint dt=" constant, " 0 lt x lt 2pi and f(pi)=2 , then the value of f(pi//2) is

Let: a_n=int_0^(pi/2)(1-sint)^nsin2tdt Then find the value of lim_(n->oo)na_n

If the value of the definite integral int_0^1(sin^(-1)sqrt(x))/(x^2-x+1)dx is (pi^2)/(sqrt(n)) (where n in N), then the value of n/(27) 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

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

Let g(x)=int(1+2cosx)/((cosx+2)^2)dxa n dg(0)=0. then the value of 8g(pi/2) is __________

Let k(x)=int((x^2+1)dx)/((x^3+3x+6)^(1/3))a n dk(-1)=1/(2^(1/3)) . Then the value of k(-2) is____

Let f be continuous and the function g is defined as g(x)=int_0^x(t^2int_0^tf(u)du)dt where f(1) = 3 . then the value of g' (1) +g''(1) is

Let f(x)=int_2^x (dt)/sqrt(1+t^4) and g be the inverse of f . Then, the value of g'(x) is

Let f(x)=int_0^x(dt)/(sqrt(1+t^3))a n dg(x) be the inverse of f(x) . Then the value of 4(g^(primeprime)(x))/((g(x))^2)i s____