Integration by substitution called Weierstrass substitution?

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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

Second image

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1 Answer

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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

$$a(1-t^2)+2bt=c(1+t^2),$$

which is quadratic.

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