Question

Solve:
$\int \frac{e^x}{\left(1+e^x\right)\left(2+e^x\right)}$ dx

Answer 1

$I=\int \frac{e^{x}dx}{(1+e^x)(2+e^x)}$

put $e^x=t$

$e^{x}dx=dt$

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

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

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

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

$I=\ln \left|c\left(\frac{t+1}{t+2}\right)\right|c=$ constant

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