A new algorithm for integral of trigonometric functions with mechanization
The authors systematically study the problem of integration of a rational expression \(R(\sin x,\cos x)\) by the substitution \(t = \tan {x \over 2}\). They observe that the resulting function \(F(\tan {x \over 2})\) has jump discontinuities at \(2k\pi + \pi\) \((k\in {\mathbb Z})\). If \(F(-)\), \(F(+)\) are the left- and right-hand limits of \(F(\tan {x\over 2})\) at \(2k\pi + \pi\) \((k\in {\mathbb Z})\), then a primitive function of \(R(\sin x, \cos x)\) reads as follows \(G(x) = F(\tan {x \over 2}) - F(-)\) for \(x \in [2k\pi,\, 2k\pi + \pi)\), \(G(2k\pi + \pi) = 0\), \(G(x)= F(\tan {x \over 2}) + F(+)\) for \(x \in (2k\pi + \pi,\, 2k\pi +2\pi]\) \((k \in {\mathbb Z})\). Further a new Maple program for the integration of \(R(\sin x, \cos x)\) is proposed.
scientific article; zbMATH DE number 2165260
scientific article