August 31st, 2019, 11:38 PM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 642 Thanks: 91  Evaluate integral
Evaluate $\displaystyle I_{x}= \int_{1}^{x} \ln \ln(t) dt$ .

September 1st, 2019, 05:25 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222 
This can't be done without resort to special functions (which wouldn't help, as you'd need a special function that is defined as an integral).

September 1st, 2019, 06:09 AM  #3 
Senior Member Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 
The Idontknow function, named for one of its proponents and denoted $\displaystyle I(x)$ (alternately, $\displaystyle I_x$), is defined as $\displaystyle I(x)= \int_{1}^{x} \ln \ln(t) dt = \int_{1}^{e^x} e^s ln(s) ds$. It was made famous by the countless variations of the inevitable humorous introduction in calculus classrooms: Professor: Can you tell me how we evaluate this integral? Student: I don't know. Professor: That's right! 
September 1st, 2019, 10:28 AM  #4 
Senior Member Joined: Dec 2015 From: somewhere Posts: 642 Thanks: 91 
I agree . $\displaystyle I_x =t\ln\left(\ln\left(t\right)\right)\operatorname{li}\left(t\right)+\lim_{x\rightarrow 1} [li(x)xlnln(x) ]$. Now only the limit is left : $\displaystyle \lim_{x\rightarrow 1} [li(x)xlnln(x)]$. 
September 1st, 2019, 11:02 PM  #5 
Senior Member Joined: Aug 2012 Posts: 2,393 Thanks: 749  I just watched Sal Khan of Khan Academy find the antiderivative of $\ln x$ using integration by parts. It's $x \ln x  x$. Might be worth playing around with except that Skipjack already said it can't be done by elementary means. How do we know that, if I may ask?

September 1st, 2019, 11:55 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222 
WA gave an answer using a special function that implies the integral can't be done, but a formal proof might be tedious, as it would rely on a formal proof that some particular special function can't be defined in closed form.


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