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?

$dx$ is called a differential and you can think of it as the "operator" $d$, known as the differential operator, acting on the expression $x$.
If you have seen the notation
$$y'(x)=\frac{dy}{dx},$$ the same operator is in use, and you are allowed to take the ratio of two differentials.
In the past, $dx$ was understood as an infinitesimally small quantity. In more modern notation, $dx$ represents the linear part of the variation of the expression that follows it. 
For example the variation around $x^2$ is $(x+h)^2-x^2=2xh+h^2$, of which the linear part is $2xh$ (when $h$ varies, $2xh$ varies proportionally, while $h^2$ varies quadratically). For this reason, we denote $dx^2=2x\ dx$.
The integration operator $\int$ can be seen as the inverse of $d$, such that
$$\int y'(x)\,dx=\int dy=y(x)+C.$$
It solves the problem "what is the function of which the differential is what follows").
Hence
$$\int 2x\,dx=\int dx^2=x^2+C.$$
The additive constant $C$ is there because the inverse is indeterminate to a constant.
As clumsy as is may appear, this notation becomes indispensable for functions of several variables, where you need to specify on which variable you differentiate/integrate.
