I encountered this expression quite a lot of times as a part of the integrating factor while solving linear differential equations.
$$e^{\int \frac {1}{x}dx}$$
For sometime, I wrote it as $x$, and was satisfied as even the answer given in my textbook had $x$ instead of $|x|$. But after realising the possibility to be $|x|$, I am confused.
What should be the answer, and why?
Edit: (My reasoning)
$\int \frac {1}{X} dx = log |x|$ and $e^{logt} = t$ if I'm not wrong. So in this case, $t = |x|$ so the answer should be $|x|$.
What is wrong with this reasoning? And could you please provide a mathematical proof if the answer should be $x$?
Edit 2:
Wolfram Alpha evaluates e^(∫(1/x)dx)
to $x$