Đặt: $t=e^x \Rightarrow dt=e^xdx$
Ta có:
$\int\limits_{\ln 3}^{\ln 5}\dfrac{dx}{e^x+2e^{-x}+3}$
$=\int\limits_{\ln 3}^{\ln5}\dfrac{e^xdx}{e^{2x}-3e^x+2}$
$=\int\limits_3^5\dfrac{dt}{t^2-3t+2}$
$=\int\limits_3^5\left(\dfrac{1}{t-2}-\dfrac{1}{t-1}\right)dt$
$=\ln\left|\dfrac{t-2}{t-1}\right|\left|\begin{array}{l}5\\3\end{array}\right.$
$=\ln\dfrac{3}{2}$