I have come across this a few times
$$\Pr(x<X)= \int_{-\infty}^x \mathrm dx' f(x')$$
This is clearly just the integral of the PDF up to the value $x,$ but there must be some very powerful reason to use that apostrophe that trump the ambiguity it introduces with the common uses of an apostrophe: derivative and transposed.
The expression $x'$ (read as '$x$ prime') is commonly used to denote 'an independent variable which is like $x$ but a bit different in some way'. It is just a name for a variable. Similarly, $\tilde{x}, \hat{x}, \check{x}, \ldots$ are often used. It is unfortunate that it is also used to denote e.g. the derivative of the object $x$ thought of as a function of some other variable.
In this case, the idea is $x \in \mathbb{R}$ is what you want to call the variable running over the range of possible values $X$ can take; but because you need such a variable to appear both as the argument of the CDF and as the integration variable when integrating the PDF, we need two different '$x$-like variables'. So one is called $x$ and one is called $x'$. This is a common notation for dummy variables in integrations like this. But call it $\hat{x}$ or $y$ or $\xi$ or whatever you fancy.
Since you ask for the "powerful reason" for doing this -- I'm not sure whether this counts as powerful, but the benefit here is that you always know what '$x$-like' variables are indexing, namely potential values of the random variable.