This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms. The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally. This work overcomes two difficulties. The first difficulty is the evaluation of singular integrals. Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains, which are solved with FFT-based finite difference solvers. The second one is the expensive computational cost for volume integrals. Despite the use of efficient interface problem solvers, the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem. The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions. Since solving boundary integral equations on a circular boundary is so accurate, one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost. Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.