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Re: Fwd: Antiderivatives and Definite Integrals


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


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