What is the "Main Value of an Integral" (denoted as $\textbf{VP}\int_{-\infty}^{\infty} \frac{\phi(x)}{x}dx$ for $f(x) = \frac{\phi(x)}{x}$)? The definition given to us is as follows (for $\text{supp }\phi \subset (-R, R)$ ):
$$\textbf{VP }\int_{-\infty}^{\infty} \frac{\phi(x)}{x}dx = \lim_{\varepsilon \rightarrow 0^+} \: (\int_{-R}^{-\varepsilon}\frac{\phi(x)}{x}dx + \int_{\varepsilon}^R \frac{\phi(x)}{x}dx)$$
This is something we call the "Main Value of an Integral" (translated directly from Polish at least, "Wartość główna całki"). Here, we take the main value of $\int_{-\infty}^{\infty} \frac{\phi(x)}{x}dx$
But what is this in the first place? How can I interpret this somehow (maybe on an example)? In what kind of topics and problems do we use this "main value of an integral"?
And most importantly: Is "main value of an integral" even the right translation (because I couldn't find anything on the Internet)?
 A: As a distribution, the Cauchy principal value satisfies the equation $x \cdot \operatorname{pv}\frac{1}{x}=1$ and is almost the only such distribution. It is also the distributional derivative of $\ln|x|.$
A: In addition to @md2perpe's accurate comments:
Yes, there is quite a bit of traditional muddle about this idea. Yes, we can "define" it... but, as you suggest, why? :)
Yes, this functional applied to a function does arise naturally in various settings. Both as a limit, and (up to constant multiple) as a Fourier transform of the rather innocent distribution the sign function. Yes, you will want to study a bit about "distributions" to be able to easily parse the explanations about the PV thing...
Indeed, a "principal value integral" is not a literal integral. Its behavior is not that of an integral... but is still sane, if one does understand "distributions".
In complex analysis, one occasionally finds a situation in which one is looking at a limit of contour integrals... where the limit would pass directly through a (for example) simple pole. Some sources pretend to "define" the integral through a pole as obtaining "half the residue". In fact, the idea that one can "define" such things into existence is inaccurate. Such a limit can be considered carefully (it's not even about "rigor", it's about getting the right answer!), and we have things like the Plemelji-Sokholski formula... (Eminently google-able.)
Yes, also, the PV integral is the unique (up to scalar multiples) functional on test functions on $\mathbb R$ that is odd and homogeneous of degree $-1$. There are various ways to prove this. A naive heuristic would make such a thing by the obvious integral... but that integral will not converge well. Certainly not absolutely, etc. Yes, it may seem that the PV convention is just one choice of many, but, in fact, we can prove uniqueness, to that particular description is necessarily equivalent to any other. :)
(That kind of uniqueness result is always very comforting to me: goofy constructions have made me uneasy... but by now I can understand that sometimes the construction (and, thus, proof of existence) is necessarily not-so-intuitive.)
