Assume that $f:[0,1] \to R$ is a smooth function. Prove that
$\lim_{n \to \infty} \int_0^1f(x)e^{inx^3}dx = 0.$
Attempt at solution: I think the solution may require the interchange of limit and integral. Have tried using the power series expansion for $e^{inx^3}$ but it didn't help. Can't seem to use Fourier inversion either.
Any advice would be much appreciated. Thanks.