Two Definite Integrals That Are Definitely (and Surprisingly!) Equal

By Shalosh B. Ekhad, Doron Zeilberger, and Wadim Zudilin

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published in The Mathematical Intelligencer ("Gems and Curiosities" column) v. 42 (2020), 10-11.

Written: Nov. 4, 2019.

Last update of this webpage: Dec. 14, 2019

    In fond memory of our good friend, Gert Almkvist (1934-2018)

We find this identity, that looks like an exercise in Calculus 1, surprising, and beautiful. We hope that you would too.

Added Nov. 7, 2019: This version corrects a typo noticed by Greg Egan.

Added Nov. 8, 2019: Watch this neat animation, by Greg Egan.

Added Nov. 11, 2019: Alin Bostan found two beautiful alternative proofs. See here and also here.

Added Nov. 11, 2019: Mikael P. Sundqvist (from Lund, where Gert Almkvist used to be) found a quick change of variable proof, alas he had to be clever. In our proof, using the Almkvist-Zeilberger algorithm, we did not have to be clever.

Added Dec. 14, 2019: Read Fernando Chamizo's insightful note,

Added Feb. 26, 2020: Alin Bostan, Fernando Chamizo, and Mikael Sundqvist combined their insights into this very interesting article.

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