Maths is an integral part of my life

Who doesn't remember the first integration formula one ever sees? \[\int u^n=\frac{u^{n+1}}{n+1}\]
Yes, for some reason letters $u$ and $v$ are famous for integration and derivation formulas. Probably somebody would complain at this stage, "what about the constant of integration?" \[\int u^n=\frac{u^{n+1}}{n+1}+K\]
Of course there should be a constant of integration! What was I thinking? But wait... there's something else missing! What am I integrating against?\[\int u^ndu=\frac{u^{n+1}}{n+1}\]
I don't pretend to explain integral calculus in this short text, there must be (literally) hundreds of text books on integral calculus. I'm just here to rant about something I don't like. I actually don't care if one uses $u$ and $v$ and other $x$ and $y$, the name of the variable is the least important thing. For formula memorising purposes I don't even see wrong in not writing down the constant of integration. However, something that really freaks me out is not writing the differential with respect to which we're integrating.

Just as in the joke just up here, my trauma becomes important when talking about multivariable calculus. It is important to know the order of integration, particularly if the region of integration is not a rectangle. \[\int_0^1\int_0^x f(x,y)dydx\neq\int_0^x\int_0^1 f(x,y)dxdy\]
This is one of the typical exercises I would ask my vector calculus students to reverse the order of integration. But Fubini's Theorem isn't just about rewriting everything in the reversed order, it is about parametrising the integrated region in such a way that the reversed order makes sense.
\[\int_0^1\int_0^x f(x,y)dydx=\int_0^1\int_y^1 f(x,y)dxdy\]
I totally understand that Fubini's Theorem might be tricky to get at the beginning. However, something that I really dislike is when something like this appears: \[\int_a^b dxf(x)\]
WTF?! I mean, yes, in the Riemann sense, if $\{a=x_0 < x_1 < \ldots < x_n=b\}$ is a partition of $[a,b]$ then the value of the integral of an integrable function $f:[a,b]\rightarrow\mathbb R$ may be calculated as
\[\int_a^b f(x)dx=\lim_{n\rightarrow\infty}\sum_{k=1}^nf(x_k)(x_k-x_{k-1})\]
which is basically the sum of some products that may be interchanged, $xy=yx$. But I will just simply say it, this:
\[\int_a^b dxf(x)\] looks horrible. Do not do it.