Recall the definition of potential energy: $$ U_x-U_{x_0} = -\int^x_{x_0}F_x(x')dx' $$
I've seen the integral definition of work, but not this - the thing I'm specifically interested in is the notation. I've never seen differential and prime notation intermingled like this. What does it mean? Would it have been possible to simply express it as:
$$ U_x-U_{x_0} = -\int^x_{x_0}F_x(x)dx $$ or is there an important reason for the given notation?
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$\begingroup$$x'$ is a so-call dummy variable. The integral could just as well have been
$$ U_x-U_{x_0} = -\int^x_{x_0}F_x(\tau)d\tau $$
without changing the result.
The variable $\tau$ is "integrated out" and the result is a function of $x$ with $x_0$ as a parameter.
See, for example, the Wolfram article for Dummy Variable:
$\endgroup$ $\begingroup$A variable that appears in a calculation only as a placeholder and which disappears completely in the final result.
What you do it's a simple change of variables. There is no change in anything, since the integration variable is mute. In this case the primed variable doesn't mean a derivative, but just as another character to be used as variable.
$\endgroup$ $\begingroup$To add to what has been said, the second way you've expressed the integral is perhaps more natural but not rigorous. What happens when you take the derivative of the RHS with respect to $x$? One is tempted to differentiate everything containing $x$, getting the wrong result, instead of applying the fundamental theorem of calculus. To be safe, we use a dummy variable $x'$
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