`F in dt h ev a l u eofint_0^1logx dxdot`

I fx+1/x=2,t h e p r i n c i p a l v a l u e o f sin^(-1)x is

f(x)=x^(3)+1-2x int_(0)^(x)g(t)dt and g(x)=x-int_(0)^(1)f(t)dt. Then yhe value of int_(0)^(1)f(t)dt lies in the interval

Let f be a non-negative function defined on the interval [0,1] . If int_0^x sqrt(1-(f'(t))^(2))dt=int_0^x f(t)dt , 0 lexle1 and f(0)=0 then the value of lim_(x to0)int_0^xf(t)/x^2 dt is

Let f be a non-negative function defined on the interval .[0,1].If int_0^x sqrt[1-(f'(t))^2].dt=int_0^x f(t).dt, 0<=x<=1 and f(0)=0,then

A derivable function f(x) satisfies the relation f(x)=int_(0)^(1)xf(t)dt+int_(0)^(x)x^(2)f(t)dt. The value of (2f'(1))/(f(1)) is

int_(0)^( If )(f(t))dt=x+int_(x)^(1)(t^(2)*f(t))dt+(pi)/(4)-1 then the value of the integral int_(-1)^(1)(f(x))dx is equal to

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