Question

Evaluate:

$\int \frac{dx}{e^x+2e^{-x}-3}dx$.

Answer 1

$I=\int \frac{dx}{e^x+2e^{-x}-3}=\int \frac{e^{x}dx}{e^{2x}-3e^x+2}$

$e^x=t\Rightarrow e^{x}dx=dt$

$I=\int \frac{dt}{t^2-3t+2}=\int \frac{dt}{t^2-2t-t+2}=\int \frac{dt}{t(t-2)-1(t-2)}$

$=\int \frac{dt}{(t-1)(t-2)}=\int dt[\frac{1}{(t-2)}-\frac{1}{(t-1)}]$

$I=\int \frac{dt}{(t-2)}-\int \frac{dt}{(t-1)}$

$I=\ln |t-2|-\ln |t-1|+\ln \left|c\right|$

$I=\ln \left|\frac{c(t-2)}{t-1}\right|=\ln \left|\frac{(e^x-2)}{(e^x-1)}\right|$