Time asymptotic behavior of integral $\int_{- \infty}^{\infty} t \cos (\alpha t) g (\alpha) d \alpha$

Consider the concrete case that $g (\alpha) = 1 / (1 + \alpha^2)$, the integration can be performed analytically (by using Wolfram Mathematica), which gives

$$\int_{- \infty}^{\infty} t \cos (\alpha t) g (\alpha) d \alpha = \int_{- \infty}^{\infty} \cos (x) g ( \frac{x}{t}) d x$$

$$= \pi \frac{e^{- t}}{t},$$ (57)

which rapid approaches zero for large $t$.