$3.$ We have discussed the op$-$amp integrator circuit in class.

$($a$)$ In the following integrator circuit, derive the transfer function between $\backslash ($ V$\_\{\backslash$text $\{$out $\}\}\backslash )$ and $\backslash ($ V$\_\{\backslash$mathrm$\{$i$\}\}\backslash ),$ if an additional resistance $\backslash ($ R$\_\{$s$\}\backslash )$ is added to the circuit.

$($b$)$ Using the transfer function, discuss$/$calculate the outputs for dc input as well as very high frequency inputs.

$($c$)$ What kind of filter characteristics would you expect from such a circuit? Qualitatively sketch the frequency response plot for this circuit.