Improper integral calculation - limit at infinity

Question

Will you please help me prove the following limit is zero ?

$$\lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \cos(x u) du. $$

Thanks in advance

Answer 1

Hint: Check Riemann-Lebesgue lemma

Answer 2

If you don't want to appeal to Riemann-Lebesgue, you could also try integrating by parts (being careful to justify the differentiability at 0 of $u\mapsto (1-{\rm e}^{-u^4})/u^2$).