we have a known probability distribution $p(t)$ with support over $t \in [t_1,t_2]$ and a known functionl form of Kernel $K(x,t) = \frac{\partial \mu(x,t)}{\partial x}$ such that the expectation value over the support vanishes, i.e.,
$$ \int_{t_1}^{t_2} K(x,t) p(t) dt = 0 $$
Clearly, after integrating out $t$, this is an equation in terms of the variable $x$. But can you somehow express $x$ in a simpler form by replacing the integral and incorporating the boundary conditions ? Preferably something of the form
$$ \xi(x,t_1,t_2) = 0 $$
This is probably too elementary, but I'm not a mathematician, and couldn't find a satisfactory way of doing it until now. If you don't feel like wasting your time writing the complete solution, some keywords to google would also be fantastic. Thanks in advance.