Question

Integrate the function: $\frac{e^x}{(1+e^x)(2+e^x)}$

Answer 1

Let $I=\int \frac{e^x}{(e^x+1)(2+e^x)}dx$
Sustituting $e^x=t$
$\Rightarrow e^x=dx=dt$
$I=\int \frac{1}{(t+1)(t+2)}dt$
$\Rightarrow I=\int \frac{(t+2)-(t+1)}{(t+1)(t+2)}dt$
$\Rightarrow I=\int \frac{1}{t+1}dt-\int \frac{1}{t+2}dt$
$\Rightarrow I=\log \left|\begin{array}{c}t+1\end{array}\right|-\log \left|\begin{array}{c}t+1\end{array}\right|+C$
$\Rightarrow I=\log \left|\begin{array}{c}\frac{t+1}{t+2}\end{array}\right|+C=\log \left|\begin{array}{c}\frac{e^x+1}{e^x+2}\end{array}\right|+C$
Where $C$ is constant of integration.