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In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general[1] transformation formula is:
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.[2] Leonhard Euler used it to evaluate the integral in his 1768 integral calculus textbook,[3] and Adrien-Marie Legendre described the general method in 1817.[4]
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name.[5] It is known in Russia as the universal trigonometric substitution,[6] and also known by variant names such as half-tangent substitution or half-angle substitution. It is sometimes misattributed as the Weierstrass substitution.[7] Michael Spivak called it the "world's sneakiest substitution".[8]
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