Can we proof without using Laguerre polynomials that $f_n(x)=O(\frac{n!}{\sqrt{n}})$ where $f_n(x)=e^{\frac{x}{2}}\int^\infty_0 e^{-t}t^{n-\frac{1}{2}}\cos(2\sqrt{xt})dt,\qquad x \geq0.$

Proof by using Laguerre polynomials;

it's easy to show that $f_n(x)=\sqrt \pi e^{-x/2} n! L^{(-1/2)}_n(x)$ and we know that $L^{(-1/2)}_n(x)=O\Big(e^{x/2}\frac{1}{\sqrt{n}}\Big) $.