I am interested in the numerical calculation of $$ F(\eta)=\frac{2}{π}\int_0^{+\infty}\sin(t\eta-t^3)\frac{dt}{t}\quad\text{for $\eta\ge 0$}. $$ I believe that the function $F$ is bounded, but I do not know its upper bound. It is easy to get $F(0)=-1/3$ and various other values for $\eta \in [0,40]$ but getting above the value $40$ for $\eta$ seems to raise computational difficulties for Mathematica. It is also interesting to see that $F$ takes values strictly above 1 with the seemingly largest value $1.5$ for $\eta =3$.

As a motivation, one may note that with $g(t)= e^{it^3}\text{pv}\frac{1}{πt}$, the operator of convolution with $g$ is bounded on $L^2(\mathbb R)$ whenever its Fourier transform is bounded. We have here $F=\hat g$, up to some normalization, and a precise bound in $L^\infty$ for $F$ would provide the $\mathcal B(L^2)$ operator-norm of that convolution.