Weierstrass substitution

In integral calculus, the Weierstrass substitution or tangent half-angle substitution is a method for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting .[1][2] No generality is lost by taking these to be rational functions of the sine and cosine. The general transformation formula is

It is named after Karl Weierstrass (1815–1897),[3][4][5] though it can be found in a book by Leonhard Euler from 1768.[6] Michael Spivak wrote that this method was the "sneakiest substitution" in the world.[7]

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