Abstract: This paper presents some examples to explain that the integral equation methods (or the integral transforms) play an important role in the astronautical mathematics.1) We employ the Fredholm integral equation methods to solve the boundary walue problem (BVP) of free vibration of stiffened ractangular plates, which is the simpler way to determine their exact natural frequencies.2) We use the techniques of Fourier and Laplace transforms to solve the BVP of unsteady transonic flows, which not only simplify the solving process but also satisfy the far field boundary automatically. Thus, a closed formula solution can be obtained.3) We employ the linearized integral theory to solve the three dimensional unsteady flow in a shear layer, which simplify the procedure analysis and produce an explicit formula for the wall pressure.4) Using the Fourier transforms and evolution integral techniques to treat the phase equations of the free electron laser, we can get a formula which describe self-consistently the nonlinear three-dimensional steady-state free electron laser amplifier.