When we work with function $f : \Bbb R \to \Bbb R$ it's common to denote the integral of $f$ over $[a,b]$ as simply:
This notation has much to do with the fact that in classical treatment, $dx$ was thought of as an "infinitesimal lenght" in the $x$ axis, $f(x)dx$ an infinitesimal area, and the integral the sum of those infinitesimal areas. This idea of infinitesimals lead to confusions, so that it was swept away so that we have today the modern and formal standard analysis. In this case, the $dx$ is usually superfluous and we write the integral simply as:
Now, for functions $f: A \subset \Bbb R^n \to \Bbb R$ we usually write the integral exactly as above or in the "classical language" simply as:
$$\int_A f = \idotsint_Af(x_1,\dots,x_n)dx_1\cdots dx_n$$
So people usually simplify writting $d^n x$ so that we have:
$$\int_A f = \idotsint_A f(x) d^nx$$
Now, this is just notation. Rigorous meaning, however can be given in the context of differential forms. In truth, when we study analysis over $\Bbb R^n$ and when we study differential geometry, we realize that the really meaningful objects to integrate are the so called differential forms. In that context, $dx$, $dy$ and etc receive a formal treatment: they are differential forms.
There we introduce the concept of a wedge product between forms denoted $\wedge$ and so people can usually write:
$$d^n x = dx^1 \wedge \cdots \wedge dx^n$$
Where $dx^i$ is the differential of the $i$-th coordinate function. It turns out that this is a "volume form", then scalar functions can be put into correspondence with $n$-forms in $n$-space, so that for $f : \Bbb R^n \to \Bbb R$ we associate the differential form $\omega \in \Omega^n(\Bbb R^n)$ given by :
$$\omega = f d^n x = f dx^1 \wedge \cdots \wedge dx^n$$
To see more about differential forms look at Spivak's Calculus on Manifolds. I hope this helps you somehow. Good luck!