Airy's integral (a.k.a. $\widehat{e^{-ip^3/3}}$ times some constant multiple) is given by $\displaystyle\text{Ai}(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ip^3/3}dp$.
Although it looks like it should diverge at $\infty$, apparently you can use integration by parts to reduce the formula to an integral with $p^3$ in the denominator plus some obviously finite terms. (I've also heard you can reduce the formula to an integral with $p^2+1$ in the denominator.) I can't figure out how to set up the by-parts to get to that reduction. Any suggestions would be greatly appreciated.
I would write it as $$ \int_{-\infty}^\infty [(p^2+x) \exp(-i(p^3/3+px))] \frac1{p^2+x} \, dp $$