I PF Integral Bee: Share Interesting/Quirky Integrals!

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The discussion invites participants to share interesting and quirky integrals, fostering a collaborative exploration of unique mathematical concepts. One integral shared is the definite integral from 0 to 1 of ln(x+1)/(x^2+1), which has sparked fascination due to its connection to the derivative of the natural logarithm. The thread emphasizes the enjoyment of discovering and understanding integrals, even for those who may not study them rigorously. Participants are encouraged to contribute their own intriguing integrals to enrich the conversation. Overall, the thread aims to create a fun and engaging environment for integral exploration.
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Thought it could be fun to have a sort of "PF Integral Bee"... if you know some interesting/quirky/etc. integrals then post them here! 🤓

To get the ball rolling...
1. ##\displaystyle{\int_0^1} \dfrac{\ln{(x+1)}}{x^2+1} dx##
 
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Not studied integrals seriously, but for me it has been really fascinating to understand ##\dfrac{d}{dx}\ln x=\dfrac{1}{x}##. The demonstration is similar to the demonstration of fundamental theorem of calculus.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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