Where does this integral for a 'aerospike nozzle' come from?
I expect you know how a conventional rocket nozzle works: It expands the exhaust gas, and the pressure inside helps to increase thrust, pushing the nozzle forward.

Now turn the conventional nozzle inside out. The expansion happens on its outside, but there is still a pressure acting on a backward-facing surface. The graph doesn't explain it, but the ramp area $\text{A}_{\text{Ramp}}$ is the projection of the nozzle area in flight direction. The ramp pressure $\text{p}_{\text{Ramp}}$ is the pressure acting along the nozzle surface, and it must be reduced by ambient pressure $\text{p}_{\infty}$ in order to arrive at the correct thrust. The integration is needed because $\text{p}_{\text{Ramp}}$ varies over the nozzle area.