Evaluate $\int_{1}^\infty \frac{\ln(\ln(x))}{1+x^2}dx$

Asked Jul 17 '18 at 00:55

Active Jul 17 '18 at 01:14

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I'm trying to evaluate: $$\int\limits_1^\infty \frac{\ln(\ln(x))}{1+x^2} dx$$

I've been told by the user Jack D'Aurizio that I can connect it to Euler's Beta Function using the substitution $x=e^u$ and using Feynman's Technique. I've tried connecting it to $$\int\limits_0^\infty \frac{t^{x-1}}{(1+t)^{x+y}}dt $$ but I don't see anything. Would appreciate a hint to put me in the right direction.

edited Jul 17 '18 at 01:01

David G. Stork

29,774

asked Jul 17 '18 at 00:55

Tom Himler

2,294

1

This is known as Vardi's integral. There are many references outside this community, and even it was dealt with in the following posting: https://math.stackexchange.com/questions/121545/evaluating-int-01-log-log-left-frac1x-right-fracdx1x2 – Sangchul Lee Jul 17 '18 at 01:21

Evaluate $\int_{1}^\infty \frac{\ln(\ln(x))}{1+x^2}dx$

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