For function $f,g$ in the Schwartz class, I want to show that $$\int_{\mathbb{R}}\left(\frac{d}{dx}f(x)+xf(x)\right)\overline{g(x)}dx=\int_{\mathbb{R}}f(x)\overline{\left(-\frac{d}{dx}g(x)+xg(x)\right)}dx$$
But I don't see how to start, since there isn't any easy way to manipulate the integrals
Notice that the term with $xf(x)\overline{g(x)}$ doesn't change on either side. Eliminating that, you're left with
$$\int_{\mathbb{R}} \frac{d}{dx}f(x)\overline{g(x)}dx = -\int_{\mathbb{R}} f(x)\frac{d}{dx}\overline{g(x)}dx.$$
If you make use of integration by parts and the fact that conjugation commutes with differentiation, you will have the result.
This kind of result is actually a pretty nifty one that gets use in quantum mechanics when dealing with quantum mechanical charge operators. The charge operators for the harmonic oscillator Hamiltonian are (up to normalization) $-\frac{d}{dx} + x$ and $\frac{d}{dx} + x$, which are adjoints of each other. You can think of your result being $\langle (-\frac{d}{dx}+x)f,g\rangle = \langle f,(\frac{d}{dx}+x)g\rangle$ (if you embed the Schwartz class into $L^2$ with the usual inner product).
