Question:
Evaluate the following integrals:
$\int \frac{e^{x}}{\left(1+e^{x}\right)^{2}} d x$
Solution:
Assume $1+\mathrm{e}^{\mathrm{x}}=\mathrm{t}$
$\Rightarrow \mathrm{d}\left(1+\mathrm{e}^{\mathrm{x}}\right)=\mathrm{dt}$
$\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$
$\therefore$ Substituting $t$ and $d t$ in given equation we get
$\Rightarrow \int \frac{1}{t^{2}} d t$
$\Rightarrow \int t^{-2} \cdot d t$
$\Rightarrow \frac{-1}{t}+c$
But $1+e^{x}=t$
$\Rightarrow \frac{-1}{1+e^{x}}+c$