When the integrator has an integrable derivative, $d\alpha$ becomes the weight $\alpha'(x)\,dx$.Suggest a slogan
Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing and differentiable with \(\alpha'\) Riemann integrable on \([a,b]\), and let \(f\colon[a,b]\to\RR\) be bounded. Then \(f\in\mathcal R(\alpha)\) on \([a,b]\) if and only if \(f\alpha'\) is Riemann integrable on \([a,b]\), and in that case \[ \int_a^b f\,d\alpha=\int_a^b f(x)\alpha'(x)\,dx. \] [@Rud76].
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P-RA-WORKSHOP-D6-10Reduction of a Stieltjes integral to a Riemann integraluses