Jim,

It can - in a sense.

And it could - in a sense - back when you were at U. of T.

The antiderivative is

(1/2) sqrt(pi) erf(-ln(a) x) / sqrt(-ln(a)) + C

HOWEVER, notice the "erf" in there. This is not an elementary function, but a function whose definition involves the integral of e^-x2, a very similar function. ("Erf" stands for "error function" and refers to its use in computing the probabilities of normally-distributed errors.) So the solution isn't really a solution, but just a statement of how it fits in with other integrals of the same type.

Thus, your instructor was "morally right" in the sense that the integral cannot be found in terms of anything familiar. This is a standard pattern: derivatives make things simpler and more elementary, do antiderivatives have to make things less elementary.

Good Hunting!
RD