An Almost Nice Integral

In using the energy method to solve (partial) differential equations, one might encounter an integral that is similar to the following: \[\frac{1}{2}\int_{a}^{b}|f’(x)|^2\;\mathrm{d}x\] where \(a,b \in \mathbb{R}\) are some constant limits of integration and \(f’(x)\) means the first derivative of a ‘nice enough’ function \(f\).

Ignoring the constant and interpreting it as a Riemann–Stieltjes integral, one may arrive at this integral that I call an almost nice integral: \[\int_{a}^{b}f’(x)\;\mathrm{d}(f(x)).\]

This almost looks like an integral that can be easily simplified using the Fundamental Theorem of Calculus (FTOC) and then evaluated if we know the definition of \(f\). Unfortunately, solving the integral above needs more work than just invoking the FTOC. We don’t even know if it actually makes sense.

If we rewrite it in a clearer way by defining \(y=f(x)\), we then obtain (with some abuse of notation): \[\int_{f(a)}^{f(b)}\frac{\mathrm{d}y}{\mathrm{d}x}\;\mathrm{d}y=\int_{f(a)}^{f(b)}\frac{1}{(f^{-1})’(y)}\;\mathrm{d}y\] where \(x=f^{-1}(y)\) is defined assuming that \(f\) satisfies the premises of the inverse function theorem.

It seems like this is as far as we can go, with a lot of strong assumptions made on the function \(f\) itself. In other words, we are unable to give a very generalised solution only based on this information. For example, consider the following integrals \[\int\left(\frac{1}{x}\right)\;\mathrm{d}(\ln x)=-\frac{1}{x}+C, \quad \text{but } \int e^{x}\;\mathrm{d}(e^x)=\frac{e^{2x}}{2}+C,\] hinting that we cannot derive a solution in terms of only \(f\), \(f’\) and \(x\).

There is actually a discussion thread about the exact same integral here on Mathematics Stack Exchange, along with another similar but more insightful post. However, the responses seem to suggest that it is impossible to arrive at a solution with then integral sign removed, without specifying what \(f\) is.

Or is it? Feel free to share your thoughts here.

An Almost Nice Integral