Let `g(x)=int_0^x f(t).dt`,where f is such that `1/2<=f(t)<=1` for `t in [0,1]` and `0<=f(t)<=1/2` for `t in [1,2]`.Then g(2) satisfies the inequality

If f(x)=x+int_0^1 t(x+t) f(t)dt, then find the value of the definite integral int_0^1 f(x)dx.

Let f(x),xgeq0, be a non-negative continuous function, and let F(x)=int_0^xf(t)dt ,xgeq0, if for some c >0,f(x)lt=cF(x) for all xgeq0, then show that f(x)=0 for all xgeq0.

Let f(x) be a continuous and differentiable function such that f(x)=int_0^xsin(t^2-t+x)dt Then prove that f^('')(x)+f(x)=cosx^2+2xsinx^2

Let f(x)=int_(2)^(x)f(t^(2)-3t+4)dt . Then

If x int_0^xsin(f(t))dt=(x+2)int_0^x tsin(f(t))dt ,w h e r ex >0, then show that f^(prime)(x)cotf(x)+3/(1+x)=0.

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_(1)^(x)(3^(t))/(1+t^(2))dt , where xgt0 , Then

Let f(x)=1/x^2 int_0^x (4t^2-2f'(t))dt then find f'(4)

If f(x)=int_(0)^(x)|t-1|dt , where 0lexle2 , then

Let f(x) be a continuous function AAx in R , except at x=0, such that g(x)= int_x^a(f(t))/t dt , prove that int_0^af(x)dx=int_0^ag(x)dx