First Differentatior and Integrator
I think I found an answer to this question. So, I will post it here. 

For power transfer function of $1-z^{-1}$ we know:
\begin{eqnarray*}
1-z^{-1}&amp;=&amp;1-e^{-jw}\\
&amp;=&amp; 1-\cos(\omega)+j\sin(w)
\end{eqnarray*}
Then by using complex conjugate properties we have:
\begin{eqnarray*}
|1-z^{-1}|^2&amp;=&amp;1-2\cos(\omega)+\cos(\omega)^2+\sin(\omega)^2\\
&amp;=&amp;2(1-\cos(\omega)).
\end{eqnarray*}
And simillarly, for $1+z^{-1}$, we have:
\begin{eqnarray*}
|1+z^{-1}|^2&amp;=&amp; 1+2\cos(\omega)+\cos(\omega)^2+\sin(\omega)^2\\
&amp;=&amp; 2(1+\cos(\omega)).
\end{eqnarray*}

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Note that the integrator in this question is a special case (see the expression for $S_i$) and is different than the conventional integrator $y[n]=y[n-1]+x[n]$.