Given ∫ eˣ ((x-1)/(x²)) dx = eˣ f(x) + c. Then f(x) satisfying the equation is:

Given $\int e^x \left(\frac{x-1}{x^2}\right) \, dx = e^x f(x) + c$. Then $f(x)$ satisfying the equation is:

(a)$x$
(b)$x^2$
(c)$\frac{1}{x}$
(d)None of the above options

Answer

Correct option: (c)

Answer: (c) $\frac{1}{x}$ $\int e^x\left(\frac{1}{x} - \frac{1}{x^2}\right)dx$ is of the form $\int e^x\,(f(x) + f'(x))\,dx$ with $f(x) = \frac{1}{x}$. Hence it equals $e^x\cdot\frac{1}{x} + c$, so $f(x) = \frac{1}{x}$.