Question

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

Answer 1

This can be solved using partial fractions and substitution.

Substitute $e^x=t.e^{x}dx=dt$

Integral becomes $\int \frac{1}{(1+t)(2+t)}dt$

Using partial fractions,

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

$\int \frac{2}{2+t}dt=2\ln |t+2|$

$\int \frac{1}{1+t}dt=\ln |t+1|$

$\int \frac{1}{(1+t)(2+t)}dt=2\ln |t+2|-\ln |t+1|+c$