Frankly I don't know how to simplify or which theorem to apply. I do know theorem on $e^{-y^2}$ (gaussian) but I don't think I can manipulate the current question to that form?
P.S Answer is 2.
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Changing the order of integration yields
$$\int_0^\infty \int_x^\infty\frac1ye^{-y/2}\,dy\,dx=\int_0^\infty \int_0^y \frac1ye^{-y/2}\,dx\,dy$$
Can you finish now?
Mark Viola
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1I think you misread the question, the power of $e$ is $y/2$ not $y^{-2}$. I – ManishKumar Singh Jan 27 '18 at 18:03
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@ManishKumarSingh I've edited. – Mark Viola Jan 27 '18 at 18:05
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@Jason I've edited. But the methodology was not impacted whatsoever. Only the argument of the exponential function. – Mark Viola Jan 27 '18 at 18:05