> I hear that in electronics and quantum dynamics, there are sometimes integrals whose value is not a number, but a function, and knowing that function is important in order to know how the thing it’s modeling behaves in interactions with other things.

I'd be interested in this. So finding classical closed form solutions is the actual thing desired there?

I think what the author was alluding to was the path integral formulation [of quantum mechanics] which was advanced in large part by Feynman.

It's not that finding closed form solutions is what matters (I don't think most path integrals would have closed form solutions), but that the integration is done over the space of functions, not over Euclidian space (or a manifold in Euclidian space, etc...)