Computing the action for a particle in a gravitational field

[AxiomOfChoice]

A friend of mine told me he fielded this at an oral exam: "Compute the classical action $S$ for a particle of mass $m$ in a gravitational field $U = -\alpha/r$." I know the formula for the classical action is given by

$$S = \int_{t_i}^{t_f} L(q,\dot q,t) dt,$$

and that for a particle in a gravitational field, we have

$$L = \frac 12 m \dot{\mathbf{r}}^2 + \frac{\alpha}{r}$$

(where, of course, $|\mathbf{r}| = r$) so that

$$S = \int_{t_i}^{t_f} \left( \frac 12 m \dot{\mathbf{r}(t)}^2 + \frac{\alpha}{r(t)} \right) dt.$$

But how in the WORLD am I supposed to perform this integration? Am I supposed to derive expressions for $\mathbf{r}$ and $r$ as functions of $t$?

 

[AxiomOfChoice]

Wow...well, I guess I don't feel so bad about not having been able to do this now!

 

[dextercioby]

You don't know who r(t) is, so you can't compute the integral, can you ? I think your last expression is exactly what the problem/examinator asked for. So A^+, huh ? :D