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](https://www.wolframalpha.com/input/?i=e%5E(%E2%88%AB(1%2Fx)dx)) evaluates `e^(∫(1/x)dx)` to $x$