Let f be a differentiable function satisfying `f(xy) =f(x).f(y).AA x gt 0, y gt 0 and f(1+x) =1+x{1 + g(x)}`, where `lim_(xto0) g(x) = 0` then `int(f(x))/(f.(x))dx` is equal to

Options

A. `(x^(2))/(2)+C`
B. `(x^(3))/(3)+C`
C. `(x^(2))/(3)+C`
D. none of these

Answer

A

Explanation

We have `f(xy)=f(x)f(y)"for all "x gt 0, y gt 0`
`Rightarrow f(x)=x^(n)`
`Rightarrow f(1+x)=(1+x)^(n)`
`Rightarrow 1+x[1+g(x)]=(1+x)^(n)`
`Rightarrow 1+x[1+g(x)]=1+nx" "[therefore (1+x)^(n)=1+nx,"if "x to 0]`
`Rightarrow 1+g(x)=n` For all x in hte neighbourhood of x=0 ltbRgt `Rightarrow g(x)=n-1` for all x in the neighbourhood of x=0
But `underset(x to 0)lim g(x)=0`
`therefore n-1=0 Rightarrow n=1`
`"Hence",f(x)=x`
`therefore int (f(x))/(f'(x))dx=int xdx=(x^(2))/(x)+C`