Question

Help with double-log integral

Answer 1

\[\huge \int_{x=1}^\infty \large{\frac{ e^{(-\frac{(\ln{(\ln{x})}-\mu)^2}{(2 \sigma^2)})}}{\sqrt{2 \pi} \sigma \ln(x)} dx}\]

Answer 2

I'm expecting something that looks similar to \[\huge e^{e^{\mu+\frac{\sigma^2}{2}}}\]

Answer 3

substitute
\[\ln(lnx)-\mu=z\]

Answer 4

\[\frac{ dx }{ lnx }=x*dz=e ^{e ^{z+u}}dz\]

Answer 5

Okay, so \[\large\frac{1}{\sqrt{2\pi}\sigma}\huge\int_{\large z=-\infty}^{\large \infty}e^{-\frac{z^2}{2\sigma^2}} \ d{z}\]

Answer 6

\[\large\frac{1}{\sqrt{2\pi}\sigma}*\frac{\sqrt{2\pi}}{\sqrt{\frac{1}{\sigma^2}}}=1\]I think I'm missing something. Wait, wouldn't \[\frac{dx}{x\ln(x)}=dz\]?

Answer 7

I see that it diverges. Rats! Now I'm looking at the interval z=-mu to +mu. Hmmm....