I am currently writing my dissertation, and there are a couple of assumptions in my work that I need to be true in order for my proofs to be valid. Without venturing into too many details, I'm currently depending on using the fact that a handful of integrals are convergent. They are quite difficult integrals, so given that I am unable to prove their convergence directly at the moment, I've made use of Mathematica. This software quickly shows that they converge to a particular number in a matter of seconds, and with a very small margin of error.
Now, I'm fully aware that this does not constitute a proof. Indeed, computers can be wrong, and should be used as a guide rather than as a substitute for a rigorous argument. My question is a very general one: when is it appropriate to use numerical evidence in this way (or in this case, a computer package which evaluates integrals numerically) as justification for a particular assumption in a proof to hold? What are examples where this would be frowned upon, and instances where this would be encouraged? Would my work be an example where such an assumption should be heavily discouraged? (I appreciate that it is difficult to pass judgement in this way without knowing the details of what my assumptions are used to prove, but I can provide details if necessary.)