Question

The first derivative of a function \( f(x) \) is given below.

\(\frac{e^x(x -1)}{x^2}\)

If f(1) = e + 3, find f(x). Show your steps.

Comments

Hii plz check my answr
It comes out to be
F(x) = e^x/x +3

Answer 1

\(f^1(x) = \frac{e^x(x -1)}{x^2}\)

\(f(x) = \int\frac{xe^x - e^x}{x^2}dx +c\)

\(= \int e^x(\frac1x - \frac1{x^2})dx +c\)

\(=\frac{e^x}{x}+C\) 

\(\left(\therefore \int e^x (f(x) + f^1(x))dx = e^xf(x) + c \right)\) 

\(\left(Here\,f(x) = \frac1x⇒f^1(x) = \frac{-1}{x^2}\right)\)

\(\because f (1) = e +3\)

⇒ \(e + 3 = e + c\)

⇒ \(c =3\)

\(\therefore f(x) = \frac{e^x}{x} +3\)

Comments

Thanks a lot for ur help
It was really helpful to me !