Evaluate the following integrals:

To evaluate the following integral following steps:

Let $e^{x}=t \ldots .(i)$

$\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}=\mathrm{dt}$

Now

$\int \frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)} d x=\int \frac{1}{(1+t)(2+t)} d t$

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

$=\log |(1+t)|-\log |(2+t)|+c$

$=\log \left|\frac{1+t}{2+t}\right|+c\left[\log m-\log n=\log \frac{m}{n}\right]$

$=\log \left|\frac{1+e^{x}}{2+e^{x}}\right|+c[$ using $(i)]$