  Much of the literature on point interactions in quantum mechanics has focused
on the differential form of Schr\"odinger's equation. This paper, in contrast,
investigates the integral form of Schr\"odinger's equation. While both forms
are known to be equivalent for smooth potentials, this is not true for
distributional potentials. Here, we assume that the potential is given by a
distribution defined on the space of discontinuous test functions.
  First, by using Schr\"odinger's integral equation, we confirm a seminal
result by Kurasov, which was originally obtained in the context of
Schr\"odinger's differential equation. This hints at a possible deeper
connection between both forms of the equation. We also sketch a generalisation
of Kurasov's result to hypersurfaces.
  Second, we derive a new closed-form solution to Schr\"odinger's integral
equation with a delta prime potential. This potential has attracted
considerable attention, including some controversy. Interestingly, the derived
propagator satisfies boundary conditions that were previously derived using
Schr\"odinger's differential equation.
  Third, we derive boundary conditions for `super-singular' potentials given by
higher-order derivatives of the delta potential. These boundary conditions
cannot be incorporated into the normal framework of self-adjoint extensions. We
show that the boundary conditions depend on the energy of the solution, and
that probability is conserved.
  This paper thereby confirms several seminal results and derives some new
ones. In sum, it shows that Schr\"odinger's integral equation is viable tool
for studying singular interactions in quantum mechanics.
