I ran into $\int\limits_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int\limits_{f(a)}^{f(b)} f^{-1}(x) dx$ proof today and was surprised to find that it has come up on the Math GRE.
I consider myself to like analysis quite a bit, so I was shocked to see this result. Are there any texts that cover this result, and on what page(s)?
A simple proof of this formula can be given based on Theorems 7.6 and 7.7 of Mathematical Analysis by Tom Apostol, concerning Riemann-Stieltjes integration. (The formula itself does not appear to be proved there, however.)
Specifically, let $f$ be a strictly monotonic bijection of an interval $[a,b]$ onto one of the intervals $[f(a),f(b)]$ or $[f(b),f(a)]$. Since $f$ is continuous, it is integrable with respect to $d\alpha$ for the integrator $\alpha(x) = x$. (This is simply a Riemann integral.) By Theorem 7.6 (integration by parts), $\alpha(x) = x$ is integrable with respect to $df$, and we have $$\int_a^b f(x) \, dx + \int_a^b x \, df(x) = bf(b) - af(a).$$
Now apply the change of variables theorem 7.7 to $\int_a^b x \, df$, taking $g$ to be the inverse function $f^{-1}$. We obtain $$\int_a^b x \, df(x) = \int_{f(a)}^{f(b)} g(x) \, d[f(g(x))] = \int_{f(a)}^{f(b)} f^{-1}(x) \, dx,$$ completing the proof.
Note Even though the end result can be stated solely in terms of ordinary Riemann integrals, if we restrict ourselves only to garden-variety substitution and integration by parts in Riemann integrals, we find that the proof requires continuous differentiability of $f$ and $f^{-1}$. The basic theorems on Stieltjes integration however yield a more general statement with no additional effort.
