May I please have help calculating the noise bandwidth for the circuit below
Firstly, it seems that your computation of the transfer function H(s) is incorrect.

Let us find the voltage across the resistor R1. Let this voltage be Va. 







simulate this circuit &ndash; Schematic created using CircuitLab

\begin{eqnarray*}
\dfrac{V_{a}}{V_{in}} &amp;=&amp; \dfrac{R{1} || (R_{2}+\dfrac{1}{sC})}{(R{1} || (R_{2}+\dfrac{1}{sC}))+\dfrac{1}{sC}} \\ &amp;=&amp; \dfrac{s(1+s)}{s^2+3s+1}
\end{eqnarray*}

wherein I assume that $$R_{1} = R_{2} = R = 1M \Omega $$ $$C=1 \mu F \hspace{2mm} such \hspace{2mm} that  \hspace{2mm} RC=1$$

Now, Va gets amplified by a gain of 100, and then Vout occurs across the final capacitor. 

\begin{eqnarray*}
\dfrac{V_{out}}{V_{a}} &amp;=&amp; \dfrac{\dfrac{100}{sC}}{R_{2}+\dfrac{1}{sC}} \\ &amp;=&amp; \dfrac{100}{1+s}
\end{eqnarray*}

\begin{eqnarray*}
H(s) = \dfrac{V_{out}}{V_{in}} &amp;=&amp; \dfrac{100}{1+s} \times \dfrac{s(1+s)}{s^2+3s+1} &amp;=&amp; \dfrac{100s}{s^2+3s+1}
\end{eqnarray*}

Moving on to the analysis of the noise bandwidth. As you mentioned, we make use of the fact that $$ s = j \omega = j 2 \pi f$$. H(f) can now be expressed as:

\begin{eqnarray*}
H(f) = \dfrac{j(200 \pi f)}{(1-4 \pi^2 f^2) +j (6 \pi f)}
\end{eqnarray*}

Taking the magnitude of this complex ratio and then squaring it, 

\begin{eqnarray*}
|H(f)|^2 &amp;=&amp; \dfrac{(200 \pi f)^2}{(1-4 \pi^2 f^2)^2 +(6 \pi f)^2} &amp;=&amp; \dfrac{(200 \pi f)^2}{16 \pi^4 f^4 + 28 \pi^2 f^2 +1}
\end{eqnarray*}

Finding Hmax can be done by simply equating the first derivative of H(f) to 0 and then solving for the value of f that yields Hmax. However, since this is rather laborious, I've taken help from Wolfram Alpha's computational knowledge engine to carry out the computation.

As for the part concerning the integral of |H(f)|^2, it can be done by partial fractions. Once again, Wolfram alpha tells us that 16 pi^4 f^4 + 28 pi^2 f^2 + 1 =0 can be factorized. 

In conclusion, the integral can be computed on Wolfram Alpha as well to give a neat answer of 0.75!

Please do edit the post if there are blatant errors!