I am trying to evalute the two following integrals
$$ I_1 = \int_0^\infty x \cos (x^3) \, \mathrm{d}x \quad \text{and} \quad I_2 = \int_0^\infty x \sin (x^3) \, \mathrm{d}x$$
I already know the numerical values, they are respectively
$$ I_1 = \frac{1}{6}\Gamma\left( \frac{2}{3} \right) \quad \text{and} \quad I_2 = \frac{1}{2\sqrt{3}}\Gamma\left( \frac{2}{3} \right) $$
Any ideas? I saw this on another forum and tried a few contours, but alas nothing worked. Thanks in advance =)