An infinite integral of cos(x^3).

Question

Is there any results about calculating this integral?? $f(\lambda )=\int_{-\infty }^{+\infty } \cos (x^3) \cos(\lambda x) \mathrm{d} x, \lambda >0.$

Answer 1

Writing $$\cos (x^3) \cos(\lambda x)=\frac 12\big(\cos (x^3+\lambda x)+\cos (x^3-\lambda x)\big)$$ There is a result which involves one of the Airy function (which is far away from elementary functions). $$f(\lambda )=\int_{-\infty }^{+\infty } \cos (x^3) \cos(\lambda x) \mathrm{d} x=\frac{\pi }{\sqrt[3]{3}}\left(\text{Ai}\left(-\frac{\lambda }{\sqrt[3]{3}}\right)+\text{Ai}\left(\frac{\lambda }{\sqrt[3]{3}}\right)\right)$$