So firstly we need to consider what mu represents, which is force of mortality at a specific age of time. Furthermore we need to consider surrounding information, which in the case of mortality models is the age of the individual (x) and the time after the age which we are interested in (t).
Thus the force of mortality becomes a function of both age (x) and time (t) and is represented by mu (x+t) and not just mu (x). To simplify this, I will put it in the context of an example, so you can see it's relevance.
Example We have a 40 year old male, and we are interested his probability of survival till age 60. Thus we set x at 40, and and t varies from 0 to 20. The answer would therefore be as follows:
$$e^{\int_{0}^{20} \mu(40+t) \space dt} $$
Alternatively we could have:
$$e^{\int_{40}^{60} \mu(x) \space dx} $$
I do hope his clarifies your issue.