`underset(x to oo)lim(overset(2x)underset(0)intte^(t^(2))dt)/(e^(4x^(2)))`equals

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

lim_(x to oo)(int_0^(2x) te^(t^(2))dt)/(e^(4x^(2))) equals

underset(n to oo)lim" " underset(r=2n+1)overset(3n)sum (n)/(r^(2)-n^(2)) is equal to

Evaluate underset(x to oo) lim ([x])/(2x)

overset(pi//2)underset(-pi//2)int (cos x)/(1+e^(x))dx=

If f(x)=overset(x)underset(0)int(sint)/(t)dt,xgt0, then

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

Consider the integrals I_(1)=overset(1)underset(0)inte^(-x)cos^(2)xdx,I_(2)=overset(1)underset(0)int e^(-x^(2))cos^(2)x dx,I_(3)=overset(1)underset(0)int e^(-x^(2//2))cos^(2)xdx and I_(4)=overset(1)underset(0)int e^(-x^(2//2))dx . The greatest of these integrals, is

The value of the integral overset(2a)underset(0)int (f(x))/(f(x)+f(2a-x))dx is equal to

lim_(x to 0)(int_(0^(x) x e^(t^(2))dt)/(1+x-e^(x)) is equal to