Đặt: $t=e^x \Rightarrow dt=e^xdx$
Ta có:
$\int\limits_1^3\dfrac{dx}{e^x-1}$
$=\int\limits_1^3\dfrac{e^xdx}{e^{2x}-e^x}$
$=\int\limits_e^{e^3}\dfrac{dt}{t^2-t}$
$=\int\limits_e^{e^3}\left(\dfrac{1}{t-1}-\dfrac{1}{t}\right)dt$
$=\ln\left|\dfrac{t-1}{t}\right|\left|\begin{array}{l}e^3\\e\end{array}\right.$
$=\ln\dfrac{e^2+e+1}{e^2}$