While reading the question about fitting all answers to the question of getting a single formula which, in the limit, captures all the answers to $\int x^{a}dx$, I thought of an obvious generalization (of the question):

What are sufficient conditions on a family of functions $f(x,a)$ so that $\int f(x,a) dx$ can be expressed as a single formula $F(x,a)+C(a)$ valid (in the limit) for all a?

From the point of view of FToC, $C(a)$ is completely irrelevant, but nevertheless such an answer has a certain elegance since $C(a)$ really captures the boundary cases of $f(x,a)$.

[Edit: although I like Dylan's answer below, it is not very algorithmic -- can one do better?]