VERY HARD Puzzle: What Is The Distance Between Two Random Points In A Square?

What is the average distance between two randomly chosen points in a square with a side length 1? The video presents how to estimate the answer and also derives the exact answer using integrals. Per suggestions in the comments, I have modified the title to indicate the difficulty level is VERY HARD (a college level calculus course and probability theory course are necessary background).
Blog post
wp.me/p6aMk-4FA
Simulation
docs.google.co...
Sources and further reading
Quadruple integral
math.stackexcha...
Difference of two uniform variables
www.math.wm.edu...
Difference of two uniform variables and solving the double integral
www.quora.com/...
Solving the double integral
math.stackexcha...
Mean distance polar coordinates change
www.mathpages.c...
Integral secant cubed
en.wikipedia.o...
Mean distance square
mathworld.wolfr...
Cleve Moler's blog post
blogs.mathwork...
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