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!
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