Write down the value of
$$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
Use the substitution \(u = x \mathrm { e } ^ { - x } + 1\) to find \(\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x\).
Hence evaluate \(\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing the limiting process used.