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Let f be a differentiable function sati...

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_(x to 0)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) implies f(1) = 1[ :. f(1) ne 0]`
Differentiating with respect of x partially , we get
`yf.(xy) = f(y) f.(x)`
Putting ` x =1 implies yf.(y) = f(y) f.(1) implies (f(y))/(f.(y)) = (y)/(f.(1))`
Now , `int(f(x))/(f.(x))dx = int(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`
`:. lim_(h to 0)(h) =0 " ":. int(f(x))/(f.(x)) dx = (x^(2))/(2) + C `
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