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Re: Integrate 5.0
*To*: mathgroup at smc.vnet.net
*Subject*: [mg44253] Re: Integrate 5.0
*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>
*Date*: Fri, 31 Oct 2003 03:01:24 -0500 (EST)
*References*: <bnnvfj$61s$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Selwyn,
> Perhaps this a better example:
>
> Integrate[Sqrt[Cos[t]^2 + 1], {t, 0, x}]
>
> For any real x, this is Sqrt[2] EllipticE[x,1/2], but here's the output
> you get:
[monstrosity snipped]
> To coax Sqrt[2] EllipticE[x,1/2] out of this mess, it's not enough to
> simplify with x\[Element]Reals,
>
> Simplify[Integrate[Sqrt[Cos[t]^2 + 1], {t, 0, x}],x\[Element]Reals]
>
> << similar output >>
>
> one has to simplify either with x\[Element]Reals&&x>0 or
> x\[Element]Reals&&x<0
>
> Simplify[Integrate[Sqrt[Cos[t]^2 + 1], {t, 0, x}],
> x\[Element]Reals&&x>0]
>
> Sqrt[2] EllipticE[x,1/2]
Thinking that perhaps it's better to let Mathematica know your intentions
during the integration, rather than afterward, I tried
Integrate[Sqrt[Cos[t]^2 + 1], {t, 0, x}, Assumptions -> x \[Element] Reals]
The result
If[x != 0, Sqrt[2]*EllipticE[x, 1/2],
Integrate[Sqrt[1 + Cos[t]^2], {t, 0, x},
Assumptions -> x \[Element] Reals && x == 0]]
while not unwieldy, is certainly strange. It's clearly equivalent to
Sqrt[2] EllipticE[x,1/2]. (So why doesn't Mathematica see that, and
simplify accordingly?)
> I'm just saying there's got to be a better way. Having a small number
> of functions that do very general things is a good philosophy, in
> general. However, in the case of Integrate, there sorely needs to be
> some simple, elegant way to integrate on the real line. But then again,
> I could be wrong.
I agree that there needs to be a way to do that.
BTW, one can throw caution to the wind and just use
Integrate[Sqrt[Cos[t]^2 + 1], {t, 0, x}, GenerateConditions -> False]
which gives the desired result immediately.
Regards,
David
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