How to convolve an arbitrary signal with a causal decaying exponential?
Indeed you have reached what can be reached, may be the following additional line can be obtained by moving the $t$ function $e^{-2t}$ out of the integral and replacing the $u(t-\tau)$ by $1$ as:

$$
y(t) = e^{-2t} \int\limits_{-\infty}^t {x(\tau)e^{2\tau}{}d\tau}
$$

In the general case you cannot simplify it any further...