Question

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

Answer 1

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

$Let\ e^x=t ⇒ e^x dx=dt$

$⇒$∫$\frac {e^x}{(1+e^x)(2+e^x)}dx$ = $∫\frac {dt}{(t+1)(t+2)}$

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

                                 = $log\ |t+1|-log\ |t+2|+C$

                                 = $log|\frac {t+1}{t+2}|+C$

                                 = $log|\frac {1+e^x}{2+e^x}|+C$