THE EVALUATION OF TRIGONOMETRIC INTEGRALS AVOIDING SPURIOUS DISCONTINUITIES

The tan(x/2) substitution, also called the Weierstrass substitution, i s one method currently used by computer-algebra systems for the evalua tion of trigonometric integrals. The method needs to be improved, beca use the expressions obtained using it sometimes contain discontinuitie s, which unnecessarily limit the domains over which the expressions ar e correct. We show that the discontinuities are spurious in the follow ing sense: Given an integrand and an expression for its antiderivative that was obtained by the Weierstrass substitution, a better expressio n can be found that is continuous on wider intervals than the first ex pression and yet is still an antiderivative of the integrand. The orig in of the discontinuities is identified, and an algorithm is presented for automatically finding the improved type of antiderivative. The ne w algorithm also enlarges the set of functions that can be used in the substitution. The algorithm works by first evaluating the given integ ral using the Weierstrass substitution in the usual way and then remov ing any spurious discontinuities present in the antiderivative.