Let $f$ and $g$ be two numerical functions such that $g(D_g)$ is contained within $D_f$ and $a$ and $b$ are two numbers of $g(D_g)$.
When trying to compute $\int_a^b fdg$, should we change the limits of the integral when expressing $g$? For instance, if we replace $g$ in the integral by its expression $g(x)$, as in, $\int f(x)dg(x)$, should we change $a$ and $b$ to $g^{-1}(a)$ and $g^{-1}(b)$, respectively?