We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by
$$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$
Show that
$$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \frac{\sqrt{n}}{2\sqrt{2}} v_{n} \quad \text{with} \quad v_{n} = \int_{0}^{\infty} \frac{1 - (\cos(\sqrt{2u/n}))^{n}}{u\sqrt{u}} \mathrm{~d}u$$