What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution?
The simplest example is the equation defining a median in statistics $\int_{-\infty}^x f(y)dy = 0.5 $, where $x$ is the unknown.
me again. I was very tired when I asked the question, the answer is trivial
$ \int_a^b f(x)dx = \Phi(b) - \Phi(a) $, where $\Phi(x) = \int f(x) dx $ is the corresponding definite integral.
So the problem boils down to being able to compute a definite integral, and then being able so solve the resulting ordinary equation.