Re: Fwd: Antiderivatives and Definite Integrals

*To*: mathgroup at smc.vnet.net*Subject*: [mg39868] Re: Fwd: Antiderivatives and Definite Integrals*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Sun, 9 Mar 2003 05:28:55 -0500 (EST)*References*: <b4c8vt$n9i$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Garry Helzer <gah at math.umd.edu> wrote: [snip] > > Redefine: > > > > In[6]:= f[x_] /; -Pi < x < Pi = f[x] > > Out[6]= 2*Sqrt[1 + Cos[x]]*Tan[x/2] > > > > In[7]:= f[x_] /; Pi < x < 3 Pi = f[x] + 4*Sqrt[2] > > Out[7]= 4*Sqrt[2] + 2*Sqrt[1 + Cos[x]]*Tan[x/2] > > Or 4Sqrt[2]Round[x/(2Pi)]+ 2*Sqrt[1 + Cos[x]]*Tan[x/2] . But these > formulas are less than perfect since they are indeterminate at odd > multiples of Pi. But of course there are "perfect" formulas for the antiderivative of Sqrt[1 + Cos[x]]. I mentioned one such formula in the parent thread of this one. If we let y denote Floor[(x+Pi)/(2*Pi)], then a "perfect" (and maximally neat?) formula for the antiderivative is 2*Sqrt[2]*( (-1)^y*Sin[x/2] + 2*y ) David Cantrell