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Re: Bug of Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82857] Re: Bug of Integrate
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Thu, 1 Nov 2007 05:19:48 -0500 (EST)
  • References: <fg4dfv$6c3$1@smc.vnet.net> <fg6pse$d44$1@smc.vnet.net> <fg9pmb$n1a$1@smc.vnet.net>

m.r at inbox.ru wrote:
> On Oct 30, 2:26 am, "David W.Cantrell" <DWCantr... at sigmaxi.net> wrote:
[snip]
> > However, related to the above, version 5.2 does give an incorrect
> > result for a definite integral with a symbolic real limit. Whether this
> > error still exists in version 6, I don't know:
> >
> > In[3]:= Assuming[Element[x,Reals],Integrate[3*Sign[Cos[t]],{t,0,x}]]
> >
> > Out[3]= 3 If[x > 0, x Abs[Cos[x]] Sec[x],
> > Integrate[Sign[Cos[t]], {t, 0, x}, Assumptions -> x <= 0]]
> >
> > The above is incorrect for x > Pi/2. A correct result would have been
> >
> > 3 Sign[Cos[x]] (x - Pi Floor[x/Pi + 1/2])
> >
> > for all real x.
> >
> > David W. Cantrell
>
> Note that your formula isn't correct for x = Pi/2 + Pi k. The correct
> expression for all real x is
>
> In[1]:= Assuming[0 <= x < 2 Pi, Integrate[3 Sign[Cos[t]], {t, 0,
> x}]] /.
>   x -> Mod[x, 2 Pi]
>
> Out[1]= Piecewise[{{-3 Pi/2, Mod[x, 2 Pi] == 3 Pi/2}, {3 (Pi - Mod[x,
> 2 Pi]), Pi/2 < Mod[x, 2 Pi] < 3 Pi/2}, {-3 (2 Pi - Mod[x, 2 Pi]), 3 Pi/
> 2 < Mod[x, 2 Pi] < 2 Pi}, {3 Mod[x, 2 Pi], 0 < Mod[x, 2 Pi] <= Pi/2}}]

Moments ago, I sent a message thanking Maxim for pointing out my error. I
also mentioned a much shorter result which is correct for all real x:

3 ArcSin[Sin[x]]

But perhaps it's worth mentioning that there is also an expression
which is correct for all real x which avoids using any functions such as
ArcSin or Sin, while still being shorter than his Piecewise expression:

3 (-1)^Floor[x/Pi + 1/2] (x - Pi Floor[x/Pi + 1/2])

That result is "in the same spirit" as what I originally intended.

David W. Cantrell


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