Let $x > 0$. Using the study of a well-chosen function, show that
$$\frac { x } { x ^ { 2 } + 1 } \varphi ( x ) \leqslant \int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t$$
where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.