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

Let $I = \int \frac{x e^{x}}{(1+x)^{2}} dx = \int e^{x} \left\{ \frac{x}{(1+x)^{2}} \right\} dx$
$= \int e^{x} \left\{ \frac{1+x-1}{(1+x)^{2}} \right\} dx$
$= \int e^{x} \left\{ \frac{1}{1+x} - \frac{1}{(1+x)^{2}} \right\} dx$
Let $f(x) = \frac{1}{1+x}$,then $f'(x) = -\frac{1}{(1+x)^{2}}$
$\Rightarrow \int \frac{x e^{x}}{(1+x)^{2}} dx = \int e^{x} \{f(x) + f'(x)\} dx$
It is known that $\int e^{x} \{f(x) + f'(x)\} dx = e^{x} f(x) + C$
$\therefore \int \frac{x e^{x}}{(1+x)^{2}} dx = \frac{e^{x}}{1+x} + C$
Where $C$ is an arbitrary constant.