$\displaystyle \int_a^t \frac{pe^{-s}}{1 - p(s - t)}ds$
Note that t and p are constants in the integral. So....
$\displaystyle \int_a^t \frac{pe^{-s}}{1 - p(s - t)}ds = p\int_a^t \frac{e^{-s}}{(1 + pt) - (ps )}ds$
To make this a bit more transparent, let 1 + pt = y, where y is of course a constant.
$\displaystyle \int_a^t \frac{pe^{-s}}{1 - p(s - t)}ds = p\int_a^t \frac{e^{-s}}{y - ps }ds$
What can you do with this?
-Dan