If `f^(prime)(x)=f(x)+int_0^1f(x)dx ,gi v e nf(0)=1,` then the value of `f((log)_e 2)` is (a)`1/(3+e)` (b) `(5-e)/(3-e)` (c)`(2+e)/(e-2)` (d) none of these

If f'(x)=f(x)+ int_(0)^(1)f(x)dx , given f(0)=1 , then find the value of f(log_(e)2) is

If f(x)=1+1/x int_1^x f(t) dt, then the value of f(e^-1) is

If f (x) =(e^(x) -e ^(-x))/( e ^(x) +e^(-x)) +2, then the value of f ^(-1) (x) is-

"If " f(x)=int(dx)/(x^(1//3)+2) " and "f(0)=12log_(e)2, " then the value of " f(-1) " is"-.

If f(x)=log_(x)(log_(e)x) , then the value of f'(e ) is -

If int (e^(x)(1+sinx))/(1+cos x)dx=e^(x)f(x)+c , then the value of f(x) is -

The value of int_1^e((tan^(-1)x)/x+(logx)/(1+x^2))dx ,is (a) tane (b) tan^(-1)e (c) tan^(-1)(1/e) (d) none of these

If f(x)=(e^(x)-e^(-x))/(e^(x)+e^(-x))+2 , then the value of f^(-1)(x) is -

If int_0^1(e^(-x)dx)/(1+e^x)=(log)_e(1+e)+k , then find the value of k.

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x) and f(1/2)=1 . Then the value of int f(x)dx lies in the interval for x:[1/2,1] (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)