Difference between these two integrators I am an electrical engineering student but new to digital signal processing. However while searching references for my senior project I came across following two different discrete integrator blocks in a MATLAB - Simulink simulation and I can't identify what these blocks do. I am familiar with continuous time integrators but this is quite new thing for me.

The above block -  Ts(z+1)/2(z-1)  is discrete time integrator with upper and lower saturation limits defined and that's all I got to know about upon clicking on it, however I want to know what does exactly that (z+1) and (z-1) means and how is it different from one shown below (KTs/(z-1)):

To the best of my knowledge above integrator is one which is direct discrete counterpart of continuous time integrator (correct me if I'm wrong), essentially Ts means it multiplies sampling time as essential in integration and 1/z-1 is inverse z-transform of a[k] = -u[-k] , right? But even if that's right, I don't understand what that term is doing here.
If my that understanding is right, what does the integrator in first block wants to tell me?
Also, the integrator in first picture is a PI torque controller, it produces reference torque command based on speed error and so that integrator is &quot;I&quot; block of &quot;PI&quot;, I think but why is it different from the one in last pic. The one in last pic just... 'integrates'...much like it's continuous time counter part.
Thanks in advance and bear with my, I'm very new to this stuff.
There are different methods to approximate integration in discrete time. The most straightforward ones are the forward and backward Euler methods, and the trapezoidal method. A discrete-time system with transfer function

$$H(z)=\frac{T}{z-1}\tag{1}$$

implements the forward Euler method.

Similarly, the backward Euler method is implemented by

$$H(z)=\frac{Tz}{z-1}\tag{1}$$

Finally, the trapezoidal method is implemented by

$$H(z)=\frac{T}{2}\frac{z+1}{z-1}\tag{1}$$

These discrete-time integrators are also described on this mathworks page.