How to convolve an arbitrary signal with a causal decaying exponential? I need to simplify the following convolution

$$x(t)\star [e^{-2t} u(t)]$$

where $u(t)$ is the unit step function. I'm very confused with this. Using the definition of convolution of continuous-time signals, I obtained

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

However, I have no idea what to do at this step. I'm stuck. Is there any property / theorem / trick that can help me simplify this?
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...