The proof of following version of the Change of Variables Theorem in Integrals is not difficult:
"Let $\phi\colon [a,b]\to [\phi(a),\phi(b)]$ be a differentiable function such that $\phi'$ is integrable and $\phi(a)< \phi(b)$ and $f\colon [\phi(a),\phi(b)]\to\mathbb{R}$ be continuous on $[\phi(a),\phi(b)]$; then $(f\circ \phi)\phi '$ is integrable on $[a,b]$ and $\int_{[a,b]}(f\circ \phi)\phi'=\int_{[\phi(a),\phi(b)]}f$"
but I'm curious about the proof of the following version, where $f$ is assumed to be just integrable and not continuous:
"$\phi:[a,b]\to [\phi(a),\phi(b)]$ differentiable monotone increasing function such that $\phi '$ is Riemann integrable; $\ f\colon [\phi(a),\phi(b)]\to\mathbb{R}$ Riemann integrable function on $[\phi(a),\phi(b)]$. Then $(f\circ\phi)\phi '\colon [a,b]\to\mathbb{R}$ is Riemann integrable on $[a,b]$, and $\int_{[a,b]}(f\circ\phi)\phi '=\int_{[\phi(a),\phi(b)]}f.$".
I read this last version on a real analysis book where the proof of this is given as a corollary of other theorems whose proofs span several pages and involve also the notion of Riemann-Stieltjes integral, so I'm wondering: does anyone know of a more elementary (i.e. not involving the notion of Riemann-Stieltjes integral but just the "usual" properties/theorems of Riemann integration)/shorter/etc. proof of this last theorem?