I have seen this recent question What's wrong in my calculation of $\int_0^{3 \pi/4} \frac{\cos x}{1 + \cos x}dx$? and I have read that when I operate for integration by substitution this type has a name: Weierstrass substitution. But is it a name for a particular substitution or applies to any substitution?

I didn't know it was called a Weierstrass substitution.

Addendum: two screenshots from two different Italian math textbooks where they write parametric formulas.

First image

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Second image

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    $\begingroup$ Weierstrass substitution means the tangent half-angle substitution. I.e., a particular kind of substitution. It doesn't mean integration by any substitution. $\endgroup$ – bjorn93 Dec 28 '19 at 23:34
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    $\begingroup$ @down-voter(s) What is the reason of the downvote to my question? Is there an exact and clear reason?!!!!!! $\endgroup$ – Sebastiano Dec 28 '19 at 23:36
  • $\begingroup$ Many books speak about Weierstrass substitution but this technique appears wall before by Euler (1707-1783) while Weierstrass (1815-1897). $\endgroup$ – Claude Leibovici Dec 29 '19 at 5:46
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    $\begingroup$ @Sebastiano: I didn't downvote, but I suspect the reason would be “This question does not show any research effort”, since the answer can be found immediately with a simple Google search. For example here: en.wikipedia.org/wiki/Tangent_half-angle_substitution $\endgroup$ – Hans Lundmark Dec 29 '19 at 9:54
  • $\begingroup$ @HansLundmark Hi, and thank you very much for further explanations. Now I put some screenshots taken from Italian textbooks. It is very simple to search on internet but it's not true that everything you read on the internet is always correct. I am not interesting for the downvote but for me it is very important to understand the reason. $\endgroup$ – Sebastiano Dec 29 '19 at 12:39

The Weierstrass substitution is precisely

$$t = \tan \dfrac{x}{2},$$ so that $$\cos x = \dfrac{1 - t^2}{1 + t^2}$$

$$\sin x = \dfrac{2t}{1 + t^2}$$ and $$dx = \dfrac{2\,dt}{1+ t^2}.$$

It rationalizes the expressions that contain trigonometric functions.

For example,

$$\int\frac{\cos x}{\cos x+1}dx=\int\frac{\dfrac{1 - t^2}{1 + t^2}}{\dfrac{1 - t^2}{1 + t^2}+1}\frac{2\,dt}{t^2+1}=\int\dfrac{1 - t^2}{1 + t^2}dt.$$

Note that Weierstrass is also useful to find the roots of trigonometric polynomials.

E.g. with the classical linear equation

$$a\cos x+b\sin x+c=0$$ we obtain


which is quadratic.

  • $\begingroup$ I call them parametric equations. when I teach goniometry to my students. But I didn't know it was called Weierstrass. $\endgroup$ – Sebastiano Dec 28 '19 at 23:38
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    $\begingroup$ @Sebastiano: "parametric equations" is a quite generic term and does not echo a substitution of a single variable. If I were you I would avoid it. $\endgroup$ – user65203 Dec 28 '19 at 23:39
  • $\begingroup$ I'll take your advice. In school textbooks they write "parametric equations in function of $t=\tan(x/2)$". $\endgroup$ – Sebastiano Dec 28 '19 at 23:41
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    $\begingroup$ @Sebastiano: they are probably referring to the fact that $(\cos x,\sin x)$ describes a circle and the Weierstrass substitution provides its parametric equations in a rational form. But this is indirect. $\endgroup$ – user65203 Dec 28 '19 at 23:44
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    $\begingroup$ @Sebastiano: there's nothing I can do about these :-) $\endgroup$ – user65203 Dec 28 '19 at 23:47

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