A general result for the integrals of the Gaussian function over the harmonic oscillator wavefunctions is derived using generating functions. Using this result, an example problem of a harmonic oscillator with various Gaussian perturbations is explored in order to compare the results of precise numerical solution, the variational method, and perturbation theory. This model problem will provide instructors of quantum chemistry with an additional example for instructional use. An additional example in which a potential of this form is used to model the ammonia vibrational spectrum is included in summary form. (Contains 3 tables.)