[agarttha]

In quantum mechanics, what you can measure experimentally (observables) are given by integrals. You can do the integrals computationally, but then you only have an empirical understanding of how the observables behave when you change some parameter of your experiment.

In our experiments, we need to know how the frequency of an electromagnetic resonator will change when we couple it to a quantum system. We calculate these frequency shifts with integrals. Being able to calculate these integrals analytically for some limiting cases helps us understand the dependence on the parameters. And usually you can patch the limiting cases together and not even have to compute the integrals numerically.