# 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, 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 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. 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, and Adrien-Marie Legendre described the general method in 1817. The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. James Stewart mentioned Karl Weierstrass (1815–1897) when discussing the substitution in his popular 1987 calculus textbook, and later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution. Michael Spivak wrote that this method was the "sneakiest substitution" in the world.