What does it mean when we write $\int\limits_a^b f(x)dg(x)$ and how to translate it into an integral with $dx$ instead?
I get intuitively that the small rate of change along the $x$ axis is no longer constant but moves in a custom way.
Does this mean that if $g$ is bijective from $[a,b]$ to $[a,b]$ we have $\int\limits_a^bf(x)dx=\int\limits_a^bf(x)dg(x)$?
$\int\limits_a^b f(x)dg(x)$ is a Riemann - Stieltjes integral
https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
For example $\int\limits_a^b f(x)dg(x)$ exists if $f$ is continuous and $g$ is of bounded variation.
If $g \in C^1[a,b]$, then $\int\limits_a^b f(x)dg(x)=\int\limits_a^b f(x)(g(x) / dx)dx = \int\limits_a^b f(x)g'(x)dx$ .