How Euler Tackled an innocent-looking integral

For unclear reasons, I really wanted to write down one wacky (read: elegant) calculus exercise tonight — have you integrated the cotangent function twice?

I feel the infinite series are mere toys dancing at the fingertips of Euler, and consider his method to tackle this quite beautiful, my usual attitude towards Euler math.

For some historical context, this integral showed up in Euler’s life-long search for the value of the inverse cubic sum of all natural numbers. He did not succeed during his lifetime, and a complete and contemporary recounting of his efforts can be read at

William Dunham (2021) Euler and the Cubic Basel Problem, The American Mathematical Monthly, 128:4,291-301, DOI: 10.1080/00029890.2021.1865014

As early-career as I am, the recently extended time I spend directly facing my inner thoughts have prompted me to imagine what it feels like to (inevitably) go into the grave (in my case be sprinkled into a black hole) with unsolved problems and unfinished work. As unfathomable the pain might be that day, I march on with a smile on my face.

To be saying this here already is, in a way, wasting that precious time.

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