# Weierstrass Substitution

The Weierstrass substitution, named after German mathematician Karl Weierstrass (1815−1897), is used for converting rational expressions of trigonometric functions into algebraic rational functions, which may be easier to integrate.

This method of integration is also called the tangent half-angle substitution as it implies the following half-angle identities:

# Tangent half-angle substitution

In integral calculus, the tangent half-angle substitution – known in Russia as the universal trigonometric substitution,[1] sometimes misattributed as the Weierstrass substitution, and also known by variant names such as half-tangent substitution or half-angle substitution – is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of x {\textstyle x} into an ordinary rational function of t {\textstyle t} by setting t = tan ⁡ x 2 {\textstyle t=\tan {\tfrac {x}{2}}} . This is the one-dimensional stereographic projection of the circle onto the line. The general[2] transformation formula is:

∫f(sin⁡x,cos⁡x)dx=∫f(2t1+t2,1−t21+t2)2dt1+t2.{\displaystyle \int f(\sin x,\cos x)\,dx=\int f{\left({\frac {2t}{1+t^{2}}},{\frac {1-t^{2}}{1+t^{2}}}\right)}{\frac {2\,dt}{1+t^{2}}}.}

The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent.[3] Leonhard Euler used it to solve the integral ∫ d x / ( a + b cos ⁡ x ) {\textstyle \int dx/(a+b\cos x)} in his 1768 integral calculus textbook,[4] and Adrien-Marie Legendre described the general method in 1817.[5] The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name.[6] James Stewart mentioned Karl Weierstrass (1815–1897) when discussing the substitution in his popular 1987 calculus textbook,[7] and later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution.[8][9] Michael Spivak wrote that this method was the "sneakiest substitution" in the world.[10]