calculus - Estimating a quotient of improper integrals

This was an extra credit question on my test on improper integrals. It was:

Define $\lfloor{x} \rfloor$ to be the greatest integer less than or equal to $x$, where $x$ is a real number. Calculate:

$$\begin{equation} \left\lfloor{\ \frac{\int_0^{\infty} e^{-x^{2}}\,dx}{\int_0^{\infty} e^{-x^{2}}\cos 2x\, dx}}\ \right\rfloor \end{equation}$$

I don't know how to start, since I don't think any of the integrals are elementary functions. Can anyone help?

Edit: He did give us the first integral as $\sqrt{\pi}/2$.

Note: This was the final $10$ point extra credit question. It was designed to be hard!