The evaluation of trigonometric integrals avoiding spurious discontinuities

Abstract

The tan(x/2) substitution, also called the Weierstrass substitution, is one method currently used by computer-algebra systems for the evaluation of trigonometric integrals. The method needs to be improved, because the expressions obtained using it sometimes contain discontinuities, which unnecessarily limit the domains over which the expressions are correct. We show that the discontinuities are spurious in the following sense: Given an integrand and an expression for its antiderivative that was obtained by the Weierstrass substition, a better expression can be found that is continuous on wider intervals than the first expression and yet is still an antiderivative of the integrand. The origin of the discontinuities is identified, and an algorithm is presented for automatically finding the improved type of antiderivative. The new algorithm also enlarges the set of functions that can be used in the substitution. The algorithm works by first evaluating the given integral using the Weierstrass substitution in the usual way and then removing any spurious discontinuities present in the antiderivative.