What change of variables is this? I have come across this integral:
$$\int \frac{f(x+tv)-f(x)}{t}g(x)dx = \int -\frac{g(x)-g(x-tv)}{t}f(x)dx$$
which the author claims is justified by a change of variables, but I cannot see what they did. Would anyone be able to elaborate?

I think he is breaking the difference apart and doing a change of variable in one term only, then recombining.
$$\int_{\mathbb R^n}\frac{f(x+tv) - f(x)}{t}\eta(x)\;dx = \int_{\mathbb R^n}\frac{f(x+tv)}{t}\eta(x)\;dx - \int_{\mathbb R^n}\frac{ f(x)}{t}\eta(x)\;dx$$
$$= \int_{\mathbb R^n}\frac{f(x)}{t}\eta(x-tv)\;dx - \int_{\mathbb R^n}\frac{f(x)}{t}\eta(x)\;dx$$
$$= \int_{\mathbb R^n}f(x)\frac{\eta(x-tv)-\eta(x)}{t}\;dx $$
