So apparently my understanding of this concept is either old, outdated, or nonstandard, etc, but I was under the assumption that in an integral, $dx$ represented the "infinitesimal change in $x$", analogous to $\Delta x$, the "width" of an approximating rectangle under the curve, where $f(x)$ is the height of that rectangle.
But now I hear it's just syntax to let you know the variable you're integrating. But if this is true then why, during things like $u$-substitution, do we still manipulate $dx$ as if it were a real quantity and we're just changing the units? It's like as if we were doing dimensional analysis for example.
So what exactly is $dx$ if it's not a real thing but is still something we apparently manipulate in certain situations such as $u$-substitution? And moreover if $dx$ is just syntax when what exactly is the integral doing if not implicitly taking the sum of infinitely many "rectangles" with infinitely small width?