Re: Integrate 5.0

*To*: mathgroup at smc.vnet.net*Subject*: [mg44284] Re: Integrate 5.0*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Tue, 4 Nov 2003 03:23:57 -0500 (EST)*Organization*: The University of Western Australia*References*: <bnnvfj$61s$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <bnnvfj$61s$1 at smc.vnet.net>, Selwyn Hollis <sh2.7183 at misspelled.erthlink.net> wrote: > I've come to the conclusion that Integrate has become nearly worthless > for computing definite integrals with symbolic limits. To cite a simple > example, > > Integrate[Sqrt[Cos[t] + 1], {t, 0, x}] > > returns an awful mess inside of an If statement (very mild in this > case) that no one should have to deal with if they're only concerned > with real numbers (specifically calculus students and a great many > applied mathematicians). I don't understand why you don't just compute the indefinite integral? The definite integration code is attempting to do a much more complicated operation. The checking code in 5.0 appears to more carefully check the conditions under which the result is true. Alternatively, why not help the integrator with appropriate assumptions, e.g. Integrate[Sqrt[Cos[t] + 1], {t, 0, x}, Assumptions -> {0 < x < Pi}] for which you get an even nicer closed form solution. > On the other hand, DSolve gives the simple, clean answer that Integrate > used to give: > > y[t]/. DSolve[{y'[t] == Sqrt[Cos[t] + 1], y[0] == 0}, y[t], t] > > 2*Sqrt[1 + Cos[t]]*Tan[t/2] But this answer is only partially incorrect. It is only valid for -Pi < t < Pi. To see what I mean, with de = {y'[t] == Sqrt[Cos[t] + 1], y[0] == 0}; compare Plot[Evaluate[y[t] /. DSolve[de, y, t]], {t, 0, 10}]; to the correct result Plot[Evaluate[y[t] /. NDSolve[{de, y, {t, 0, 10}]], {t, 0, 10}]; Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul