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    Half-angle Substitution

    Alternate names

    half-tangent substitution | half-angle substitution | tangent half-angle substitution | universal trigonometric substitution

    Method

    Let f(x) be a real-valued rational function of trigonometric fiunctions, i.e. f(x) = P(x)/Q(x) for all real numbers x, where both P(x) and Q(x) are polynomials in sin(x) and/or cos(x). Then by setting t = tan(x/2), the following half-angle formulas can be used to transform the integrand f(x) in an integral integral f(x) dx into a rational polynomial in t: sin(x) = (2 tan(x/2))/(1 + tan^2(x/2)) = (2 t)/(1 + t^2) cos(x) = (1 - tan^2(x/2))/(1 + tan^2(x/2)) = (1 - t^2)/(1 + t^2) tan(x) = (2tan(x/2))/(1 - tan^2(x/2)) = (2t)/(1 - t^2) cot(x) = (1 - tan^2(x/2))/(2tan(x/2)) = (1 - t^2)/(2t) sec(x) = (1 + tan^2(x/2))/(1 - tan^2(x/2)) = (1 + t^2)/(1 - t^2) csc(x) = (1 + tan^2(x/2))/(2 tan(x/2)) = (1 + t^2)/(2t). When combined with the corresponding transformation of the differential dx = d (2 tan^(-1)(t)) = (2 dt)/(1 + t^2) an integral of a rational expression of trigonometric functions can therefore be reduced to an integral of a rational function using the Weierstrass substitution.

    Details

    indefinite integral

    integration by substitution | trigonometric substitution

    Associated people

    Karl Weierstrass | Isaac Newton | Isaac Barrow | Gottfried Leibniz

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