Is there a process for deriving special cases of convolution?
Well, indeed there are special cases for convolutions but yours is quite straightforward. You have to consider two cases: $t&lt;0$ and $t&gt;0$. 

If you sketch the convolution process for these two cases, by sliding your time shifted and time reversed input on the same axis as your impulse response, you will end up with the following:



Can you handle the integrals shown?