Parenthesis in integral necessary? Im arguing with some friends whether it is necessary to put parenthesis around a sum in an integral: can we write $\int_a^b x^2+x ~ \mathrm d x$ or do we need to write $\int_a^b(x^2+x)\mathrm d x$?
My friends are arguing that we are not integrating $f$ but rather the differential form $f\cdot \mathrm d x$. Since this is a product we need parenthesis if $f$ is replaced by a sum.
My argument is, that i see the integral as an operator $\int_a^b \cdot \mathrm dx: L^1\to\mathbb C$, so the full notation would be $\int_a^b (x\mapsto x^2+x)dx$. Both of the suggestions above would then just be shortcuts to save some time and the necessity of paranthesis would then just be an arbitrary cut of how much of a shortcut is allowed.
I think both arguments do have some validation. 
So my questions: Do we need them or don't we?
 A: I think the "need parentheses for sum because $f~dx$ is a product" argument, though noble, fails for two $\displaystyle\lim_{\epsilon\rightarrow0}\epsilon+1$ reasons:


*

*The 'factors' in the product aren't interchangeable: you can't write $\displaystyle\int~dx~f(x)$ ... well, you can, but nobody possibly only physicists :-) will understand what you mean.

*With double and triple integrals, the integrals and differentials behave like parentheses, e.g. $\displaystyle\int_0^2\int_0^{2\pi}r^2\theta+\theta~d\theta~dr$ is effectively
$\displaystyle\int_0^2\left(\int_0^{2\pi}\left(\vphantom{\int}r^2\theta+\theta\right)~d\theta\right)~dr$. You can't reorder the differentials without changing the meaning.


Edit: physicists, at least, do write multiple integrals with leading differentials (see comments below); it's possible to assign a well-defined meaning to integrals written this way and which would require the trailing expression to be parenthesised if it were a sum rather than a product - c.f. $\cos2\theta$ vs. $\cos(2\theta+\pi)$. That style treats $\displaystyle\int~dx$ like a function or operator, as per the OP's suggestion, and does not treat the differential as a factor in a product as put forth by the OP's friends (the placement of the differentials matters as it would not in a true product).

