Prerequisites for audio signal processing
DSP is indeed a very mathematical discipline, but not completely. The amount of mathematical knowledge you need to operate DSP concepts comfortably is luckily limited to a particular fairly small subset. I would say that you need the following:


Complex variables. DSP deals with oscillating signals and systems, and those are very conveniently represented using Euler's complex exponential formulas. This includes knowledge of polynomials over complex variables, and so on.
Simple differential calculus. I would say that you need to know how to take derivatives of polynomials and other common functions such as exponentials and logarithms. Knowing l'Hôpital's rule also occasionally comes in handy. Taking derivatives is primarily required for finding minima and maxima of functions. Note that you need to be able to do this for complex variables as well, but at this stage there is practically no difference in concepts between real and complex differentiation.
Linear algebra. I put linear algebra before integration because it is much more useful, and since integration is a linear operation, a lot of it can be replaced by linear algebra concepts. I can't exaggerate how important linear algebra is, so go and learn it. I highly recommend Gilbert's MIT Open Courseware linear algebra class. No one teaches it better than this guy. Just watch the lectures.
Integral calculus. I'm not quite sure how to approach describing the importance of integration in DSP. If linear algebra was a moderately easy, fun and useful topic to learn, integration, especially complex integration, takes a lot of commitment and intuition, both of which require lots of time. Even though I usually advocate for in-depth mathematical knowledge, here's what you can very easily get away with. Learn about integration concepts, what it means and how integration is related to derivatives. Since most integrals are difficult to work out by hand, people use integration tables to look up answers instead. This will get you through most integration you need to know for DSP.
Infinite series. This is the one part of calculus that everyone forgets the minute they take that final exam, but it actually comes in handy in DSP. Specifically, when analyzing recursive systems, i.e. those that take their output and reuse it as input creating a feedback loop, you need to be able to spot several types of infinite number patterns that appear and point to a single formula that produces every single term in that infinite string of numbers or polynomial coefficients.
Probability. I hesitate a little to write about this one, because not all DSP courses strictly require probability, but if the one you're looking at does, it wouldn't hurt to know it. Not be be lost, you need to know expressions and concepts of means and variances, as well as Bayes' rule.


Obviously, what I've given above are the minimal requirements, but you may need to learn more as you go.