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The universal trigonometric substitution converts a rational function in $\sin(x), \cos(x) $ into a rational function of a new variable $t$ by the substitution $t = \tan(x/2) $. It therefore enables to solve the integral of any function in $R[\sin x,\cos x] $ by the methods of decomposition into partial fractions.

According to Wikipedia, this method is named after Weierstrass although It was used much earlier by Euler. It seems to me very elementary, and even Euler himself calculated far more difficult integrals, so i'd like to know why the name of a mathematician as modern as Weierstrass is attached to such an elementary technique; did he contribute something to it? did he just popularize it?

José Carlos Santos
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user2554
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    Boyer's law ("Mathematical formulas and theorems are usually not named after their original discoverers")? – njuffa Jul 22 '20 at 20:45
  • O.k that explains why it's not named after Euler, but among all later mathematicians why Weierstrass? – user2554 Jul 22 '20 at 21:20
  • So far, I have not been able to find any publication in English prior to 1990 that uses the specific term "Weierstrass substitution". – njuffa Jul 22 '20 at 21:28
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    French Wikipedia: "L'utilisation de cette substitution pour le calcul de primitives est parfois appelée substitution de Weierstrass, du nom de Karl Weierstrass, sans pour autant justifier cette appellation, d'autant que la technique était déjà utilisée par Leonhard Euler (1707-1783), donc avant la naissance de Weierstrass." In English: " ... named after Karl Weierstrass, without however justifying this name, especially since the technique was already used by Leonhard Euler ..." – njuffa Jul 22 '20 at 21:45
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    After hours of browsing, I have yet to find anything in the scans of the collected works of Weierstrass at the Internet Archive that would motivate the naming of the Weierstrass substitution. This terminology seems to go back to James Stewart's calculus textbooks, but I am by no means sure. – njuffa Jul 22 '20 at 23:07

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