I'm reading through the lecture notes of Wayne Hu regarding the Damping Scale of the CMB. He give the following steps to calculating the damping scale, $k_D$:$$k_D^{-2}=\int \frac{1}{6(1+R)}\left( \frac {16}{15}+ \frac{R^2}{(1+R)}\right)\frac{1}{\dot\tau} d\eta$$Limiting forms:$$\lim\limits_{R \to 0}k_D^{-2}= \frac{1}{6}\frac{16}{15}\int \frac{1}{\dot\tau} d\eta$$ $$\lim\limits_{R \to \infty}k_D^{-2}= \frac{1}{6}\int \frac{1}{\dot\tau} d\eta$$and finally$$k_D=\frac{\sqrt{6}}{\sqrt{\eta \dot\tau^{-1}}}$$I see roughly what he's doing, but my math is rusty. Could anyone explain in greater detail how he arrived at $k_D$? For example, why did he discard the limit as ${R \to 0}$? Why is only ${R \to \infty}$ used in the final formula? Why was he able to extract everything but $\frac{1}{\dot\tau}$ from the integral (since R is also a function of $\eta$)?
EDIT: While function $f(\eta)=\frac{1}{6(1+R(\eta))}\left( \frac {16}{15}+ \frac{R(\eta)^2}{(1+R(\eta))}\right)$ is relatively constant, it does change with $\eta$ and looks like this:
and the function $g(\eta)=\frac{1}{\dot\tau(\eta)}$ looks like this:
The bottom axis is $\eta$ in seconds, $s$.
From the second limiting form $$\dfrac{1}{k_D^2}=\dfrac{1}{6}\int \dfrac{1}{\tau} d\eta \\ 1=\dfrac{1}{6}\int \dfrac{1}{\tau} d\eta \ k_D^2 \\ 1=\dfrac{1}{6} \tau^{-1}\eta \ k_D^2 \\ k_D=\sqrt{\dfrac{6}{\tau^{-1}\eta}} \\ k_D=\dfrac{\sqrt{6}}{\sqrt{\tau^{-1}\eta}} $$ And as you can see, this is the answer.