Show that

$$\int_{0}^{1} e^{i k t^{3}} d t=I_{1}-I_{2}, \quad k>0$$

where $I_{1}$ is an integral from 0 to $\infty$ along the line $\arg (z)=\frac{\pi}{6}$ and $I_{2}$ is an integral from 1 to $\infty$ along a steepest-descent contour $C$ which you should determine.

By employing in the integrals $I_{1}$ and $I_{2}$ the changes of variables $u=-i z^{3}$ and $u=-i\left(z^{3}-1\right)$, respectively, compute the first two terms of the large $k$ asymptotic expansion of the integral above.