I came across the following integral and its evaluation in a scientific paper on sediment transport (https://doi.org/10.1061/(ASCE)0733-9429(2000)126:11(818) if you're interested) and I verified it in wolfram alpha, but I'm wondering if anyone can help me understand why it's true. Thanks!
By the Leibniz integral rule, we should have
$$\frac{\partial}{\partial x}\int_{-\infty}^x f(y) \left(\int_{x - y}^\infty g(t) dt\right) dy = \left[f(y)\left(\int_{x - y}^\infty g(t) dt\right)\right]_{y = x} + \int_{-\infty}^x \frac{\partial}{\partial x} \left[f(y) \left(\int_{x - y}^\infty g(t) dt\right)\right] dy$$
Evaluating the first term and reusing the Leibniz rule on the second term yields
$$f(y)\int^\infty_0 g(t) dt + \int_{-\infty}^x f(y) (-g(x - y)) dy$$
which matches your right-hand side.