So, an integral is notated like this:
And from my understanding, it's an operator that is defined for three operands: $a$ and $b$, which can be anything, and an integrand of the form $f(x)dx$.
$dx$ is just an infinitesimal number, so $f(x)dx$ is simply $f(x)$ multiplied by $dx$. This gives an infinitesimal based on $x$, and the integral is the infinite sum of all of those. I found a good page that explains this idea nicely.
But what about integrands that are not of the form $f(x)dx$? For example, things like:
Certainly, if the intuition explained above is true (i.e: $f(x)dx$ is simply an algebraic expression; therefore the integral operator must accept any algebraic expression as its integrand), then these things must be syntactically legal. Is that wrong? How should these things be interpreted?
I suspect that my interpretation of $f(x)dx$ is wrong. $dx$ is part of the integral's notation, and plays a special role in what the integral operator does (defining the variable of integration). But $dx$ is also supposed to represent a numerical value which is multiplied by $f(x)$. How can it be both of those things at once?
I'm probably overthinking this. I just want to understand the notation and the intuition behind it. I hope someone can recognize what my confusion is and rectify it for me.