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== Euler's second proof ==
The first accepted 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=Journ. lit. d'Allemange, de Suisse et du Nord|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>, 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 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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