If `f^(prime)(x)=f(x)+int_0^1f(x)dx ,gi v e nf(0)=1,` then the value of `f((log)_2 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 the value of f(log_(e)2) is

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

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

f(x)=int_1^x(tan^(-1)(t))/t dt , x in R^+, then find the value of f(e^2)-f(1/(e^2))

If int_0^1e^-(x^2)dx=a , then find the value of int_0^1x^2e^-(x^2)dx in terms of a .

If ("lim")_(tvecx)(e^tf(x)-e^xf(t))/((t-x)(f(x))^2)=2a n df(0)=1/2, then find the value of f^(prime)(0)dot a) 4 (b) 2 (c) 0 (d) 1

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)a n df(1/2)=1 . Then the value of int_(1/2)^1f(x)dx lies in the interval (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)

The domain of f(x)i s(0,1)dot Then the domain of (f(e^x)+f(1n|x|) is (a) (-1, e) (b) (1, e) (c) (-e ,-1) (d) (-e ,1)

If f(x)=(log)_e((x^2+e)/(x^2+1)) , then the range of f(x)

If f(x) =(e^x)/(1+e^x), I_1=int(f(-a))^(f(a)) xg(x(1-x)dx, and I_2=int_(f(-a))^(f(a)) g(x(1-x))dx, then the value of (I_2)/(I_1) is (a) -1 (b) -2 (c) 2 (d) 1