What is $dx$ on an integral? I've heard from some of my teachers it's a bilineal form and some other stuff, nobody actually ever explained me the reason of it. Of course i've done practical problems in which $dx$ is a "very small part of somethign", but what can we say about it in some "higher grade" notes?

dx on an integral comes (at least) from the definition of Stieltjes integral. Here the definition for real functions:
 Let $f, x:[a,b]\longrightarrow\mathbb{R}$ be real functions defined on the interval $[a,b]$. Consider a partition of the interval $[a,b]$, $P^{*} = (P,\xi)$, where:
$$P = \lbrace a = t_{0} < t_{1} < \ldots < t_{k}=b\rbrace, \ \ t_{i-1}\leqslant \xi_{i} \leqslant t_{i}.$$
The Stieltjes sum associated to this partition and the functions $f$ and $x$  is defined by:
$$\Sigma(f,x,P {*}) = \Sigma(P {*}) = \sum_{i=1}^{k}f(\xi_{i})[x(t_{i}) - x(t_{i-1})].$$
Then, (here is the answer), the Stieltjes integral of $f$ in relation to $x$ is given by:
$$\int_{a}^{b}f(t)dx = \displaystyle \lim_{\vert x\vert\rightarrow 0}\sum(P^{*}).$$
Observe that, the left rand of the equation is just a notation. 
