Question

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

Answer 1

$\frac{e^x}{(1+e^x)(2+e^x)}$
Let $e^x=t\Rightarrow e^{x}dx=dt$
$\Rightarrow \int \frac{e^x}{(1+e^x)(2+e^x)}dx=\int \frac{dt}{(t+1)(t+2)}$
$=\int [\frac{1}{(t+1)}-\frac{1}{(t+2)}]dt$
$=\log |t+1|-\log |t+2|+C$
$=\log \left|\frac{t+1}{t+2}\right|+C$
$=\log \left|\frac{1+e^x}{2+e^x}\right|+C$