Given a function `f:[0,4]toR` is differentiable ,then prove that for some `alpha,beta in (0,2), int_(0)^(4)f(t)dt=2alphaf(alpha^(2))+2betaf(beta^(2))`.

If the function f:[0,4]vecR is differentiable, the show that for a , b , in (0,4),f^2(4))-(f^2(0))=8f^(prime)(a)f(b)

Let f(x) be a differentiable function in the interval (0, 2) then the value of int_(0)^(2)f(x)dx

Let f:R to R be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . f(x) increases for

If 2 tan beta+cot beta= tan alpha , prove that cot beta= 2 tan (alpha-beta) .

Let f be a differentiable function such that f'(x) = f(x) + int_(0)^(2) f(x) dx and f(0) = (4-e^(2))/(3) . Find f(x) .

Let y = f(x), f : R ->R be an odd differentiable function such that f'''(x)>0 and g(alpha,beta)=sin^8alpha+cos^8beta+2-4sin^2alpha cos^2 beta If f''(g(alpha, beta))=0 then sin^2alpha+sin^2beta is equal to

Prove that int_(0)^(x)e^(xt)e^(-t^(2))dt=e^(x^(2)/4)int_(0)^(x)e^(-t^(2)/4)dt .

if alpha and beta are the root of the quadratic polynomial f(x) = x^2-5x+6 , find the value of (alpha^2 beta + beta^2 alpha)

Let f(x) be a differentiable function such that f(x)=x^2 +int_0^x e^-t f(x-t) dt then int_0^1 f(x) dx=

Let f:[0,2]vecR be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1 L e t :F(x)=int_0^(x^2)f(sqrt(t))dt for x in [0,2] . If F^(prime)(x)=f^(prime)(x) . for all x in (0,2), then F(2) equals