I'm interested in knowing what is the utility of using dummy variables in definite integrals; that is, why use $\int_0^xf(x')dx'$ instead of $\int_0^xf(x)dx$.
In case they have no real utility and it is only a convention used for historical reasons, it would be interesting to know what were the reasons that originally led to the introduction of this concept.
Note: I have seen this question or a similar one posted here, but I'm not sure the OP was asking the question in the sense I ask, and I don't think any of the answers given explains the matter...
It is a matter of correctness and to ease readability. In fact the term bound variable is much more appropriate than the term dummy variable since it addresses the scope of variables which is relevant here.
A discrete pendant might help to better see the situation:
Consider the formulas \begin{align*} \sum_{k=1}^n k=\frac{n(n+1)}{2}\qquad\qquad\qquad \sum_{k=1}^k k\stackrel{???}{=}\frac{k(k+1)}{2} \end{align*}
The index $k$ in the left sum is a bound variable with scope given by the sigma symbol. The variable $n$ on the other hand is a free variable whose existence is not bound by the sigma symbol. The right sum is nonsense. Free and bound variables are the main theme here and also in OPs example.