高等数学积分万能公式 Weierstrass Substitution 正切半角公式

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 ${\textstyle x}$ into an ordinary rational function of ${\textstyle t}$ by setting ${\textstyle t=\tan {\tfrac {x}{2}}}$ . This is the one-dimensional stereographic projection of the circle onto the line. The general^[2] transformation formula is:

\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 ${\textstyle \int dx/(a+b\cos x)}$ 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]