Let f be a positive differentiable function defined on `(0,oo)` and `phi(x)=lim_(nrarroo) (f(x+(1)/(n))/f(x))^(n)`. Then `intlog_(e)(phi(x))dx=`

Let f(x) be a non-constant twice differentiable function defined on (oo,oo) such that f(x)=f(1-x) and f'(1/4)=0^(@). Then

Let f (x) be a twice differentiable function defined on (-oo,oo) such that f (x) =f (2-x)and f '((1)/(2 )) =f' ((1)/(4))=0. Then int _(-1) ^(1) f'(1+ x ) x ^(2) e ^(x ^(2))dx is equal to :

Let f(x)=lim_(n rarr oo)(cos x)/((1+tan^(-1)x)^(n)), then int_(0)^(oo)f(x)dx=

Let f(x) be a non-constant twice differentiable function defined on (oo, oo) such that f(x) = f(1-x) and f"(1/4) = 0 . Then

Let f(x) be a non-constant twice differentiable function defined on (oo, oo) such that f(x) = f(1-x) and f"(1/4) = 0 . Then

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then

If phi(x)=lim_(nrarrinfty)(x^(2n)f(x)+g(x))/(1+x^(2n)) , then