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Let f be a differentiable function satis...

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

A

`(x^(2))/(2)+C`

B

`(x^(3))/(2)+C`

C

`(x^(3))/(3)+C`

D

`(x^(2))/(3)+C`

Text Solution

Verified by Experts

The correct Answer is:
A
Put `x=y=1`, we get `f(1)=f^(2)(1) rArr f(1)=1[because f(1) ne 0]`
Differentiating with respect to x partially, we get
`yf'(xy)=f(y)f'(x)`
Putting `x=1 rArr yf'(y)=f(y)f'(1) rArr (f(y))/(f'(y))=(y)/(f'(1))`
Now, `int (f(x))/(f'(x))dx=int (x)/(f'(1))dx=(1)/(f'(1))((x^(2))/(2)+c)`
`f'(1)=lim_(h to 0) (f(1+h)-f(1))/(h)`
`=lim_(h to 0) (1+h+hg(h)-1)/(h) = lim_(h to 0)1+g(h)=1`
`because lim_(h to 0)g(h)=0 therefore int (f(x))/(f'(x))dx=(x^(2))/(2)+C`
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