Do integration/differentiation processes work as simple filters?
Yes, integration and differentiation can be linear filters.
You can start from laplace properties that say:

$ \int_{0}^{t} {x(t)dt} \longrightarrow \frac{X(s)}{s} \\ \frac{d}{dt}x(t) \longrightarrow sX(s) $

So you can find transfer function of integration and differentiation:

$ H_{INT}(s) = \frac{1}{s} \\ H_{DIFF}(s)=s $

You can convert these transfer function in digital IIR filters for example by bilinear transform or others digitization techniques. However you should notice that $ H_{DIFF}(s) $ is not causal, then you must add a pole to the transfer faction far from useful signal frequency, and it begin:

$ H_{DIFF_{causal}}(s)= \frac{s}{\alpha s + 1}$ where $\alpha$ is the time constant of the derivative filter a small real $&gt;0$.

Using bilinear transform $H_{INT}(z)$ begin a trapezoidal integrator, using Euler transofrm $H_{INT}(z)$ begin a rectangular integrator, you can see this difference and the digitization in the MATLAB PID page.

I wrote a simple PID program in C that computes these operations using Euler transform, a PID in closed loop doesn't need to be accurate so Euler works well.

In literature there are a lot of ways to implement derivative filters (IIR and FIR) that workaround in elegant way anti-causal problem, however in a lot of situations you can simply digitize the analog transfer functions to make IIRs and FIRs.