Re: Integrate 5.0

*To*: mathgroup at smc.vnet.net*Subject*: [mg44242] Re: [mg44233] Integrate 5.0*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Fri, 31 Oct 2003 03:01:07 -0500 (EST)*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst*References*: <200310290834.DAA05970@smc.vnet.net>*Reply-to*: murray at math.umass.edu*Sender*: owner-wri-mathgroup at wolfram.com

I tried your integral and saw the same quite complicated "If" result. So I did what seemed the obvious thing (with output here converted to InputForm): Integrate[Sqrt[Cos[t] + 1], {t, 0, x}, Assumptions -> x ? Reals] If[x <= Pi && Pi + x >= 0, 2*Sqrt[1 + Cos[x]]* Tan[x/2], Integrate[Sqrt[1 + Cos[t]], {t, 0, x}, Assumptions -> x ? Reals && (x > Pi || Pi + x < 0)]] That doesn't seem so awful to me. Selwyn Hollis 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). > > 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] > > Could it be that we need a new function such as this: > > RealIntegral[expr_,{x_,a_,b_}]:= > (y[x]/. First@DSolve[{y'[x] ==expr, y[a] == 0}, y[t], t])/.x->b > > that would be associated with \[Integral] ? ... leaving the current > Integrate to be associated with \[ContourIntegral]?? > > Or perhaps a simple option for Integrate like RealLimits->True? > > ----- > Selwyn Hollis > http://www.math.armstrong.edu/faculty/hollis > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305