Purpose of Semidefinite Integral
It is just an opaque way to denote that the non-definite integration limit will take the value of the variable itself. Namely,

$$\int_a g(x) dx \equiv \int_a^x g(s)ds$$

... where $s$ is just the dummy variable of integration. Since sometimes we use the notation $\int _D f(x) dx$ to denote fully definite integration "over the domain $D$ of $x$", I would avoid the left-hand side notation and I would stick with the more clear right-hand-side one. 

I do hope that the authors where you find this notation mean what I wrote above, because this is what mathematicians mean by it.