Am I missing something? Why wouldn't you use and
The cdf defined as
In this case, you are integrating a piecewise function, which hopefully you know how to do - presumably your cdf will also be piecewise. Incidentally, you should note that your pdf is always going to be defined over the entire real line, so this integral always makes sense; if it seems like it isn't defined on a particular subset of the real numbers, it is taken to be identically 0 there.