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=`

Options

A. `(1)/(2)(f(x))^(2)+c`
B. `f(x).f'(x)+c`
C. `log_(e)(f(x))+c`
D. `log_(e)((f'(x))/(f(x)))+c`

Answer

C

Explanation

`phi(x)=underset(nrarroo)lim(f(x+(1)/(n))/(f(x)))^(n)=underset(hrarr0)lim(f(x+h)/f(x))^(1//h)implieslog_(e)phi(x)=underset(hrarr0)lim(log_(e)f(x+h)-log_(e)f(x))/(h)`
`implieslog_(e)phi(x)=(d)/(dx)(log_(e)f(x))impliesintlog_(e)phi(x)dx=log_(e)f(x)+c`