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== Euler's second proof ==
The first accepted and definitive proof of the theorem was made by Euler in 1741<ref>{{Cite book|title=Opera Omnia, series 1, volume 14|last=Euler|first=Leonhard|publisher=|year=1741|isbn=|___location=|pages=177-186}}</ref><ref>{{Cite book|title=Journal littéraire d'Allemange, de Suisse et du Nord, article E063|last=Euler|first=Leonhard|publisher=|year=1743|isbn=|___location=|pages=p. 115-127}}</ref><ref>{{Cite web|url=http://eulerarchive.maa.org/hedi/HEDI-2004-03.pdf|title=How Euler did it|last=Sandifer|first=Ed|date=March 2004|website=MAA Online|url-status=live|archive-url=http://archive.wikiwix.com/cache/?url=http%3A%2F%2Feulerarchive.maa.org%2Fhedi%2FHEDI-2004-03.pdf|archive-date=|access-date=}}</ref><ref>{{Cite web|url=https://www.apmep.fr/IMG/pdf/Article_probleme_Bale.pdf|title=Euler and Basel problem (page 17-19)|last=Association of Professors of mathematics of public education|first=Apmep (France)|date=|website=apmep.fr|url-status=live|archive-url=|archive-date=|access-date=}}</ref>, six years after his first proof. The rigor of the latter was challenged at that time because the Weierstrass factorization theorem had not been discovered yet. The following proof is almost the same as Euler's second proof. The integration by substitution <math>\ u = sin(t)</math> links the two proofs.
First of all, <math>\displaystyle \int ^{\frac{\pi }{2}}_{0} \operatorname{Arcsin}( \sin\ t) \ dt\ =\int ^{\frac{\pi }{2}}_{0} t\ dt\
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