       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:= f[x_] /; -Pi < x < Pi = f[x]
> > Out= 2*Sqrt[1 + Cos[x]]*Tan[x/2]
> >
> > In:= f[x_] /; Pi < x < 3 Pi = f[x] + 4*Sqrt
> > Out= 4*Sqrt + 2*Sqrt[1 + Cos[x]]*Tan[x/2]
>
> Or  4SqrtRound[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*( (-1)^y*Sin[x/2] + 2*y )

David Cantrell

```

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