Re: Integrate 5.0
- To: mathgroup at smc.vnet.net
- Subject: [mg44246] Re: [mg44233] Integrate 5.0
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Fri, 31 Oct 2003 03:01:12 -0500 (EST)
- References: <200310290834.DAA05970@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 29 Oct 2003, at 17:34, Selwyn Hollis wrote:
> I've come to the conclusion that Integrate has become nearly worthless
> for computing definite integrals with symbolic limits. To cite a simple
> example,
>
> Integrate[Sqrt[Cos[t] + 1], {t, 0, x}]
>
> returns an awful mess inside of an If statement (very mild in this
> case) that no one should have to deal with if they're only concerned
> with real numbers (specifically calculus students and a great many
> applied mathematicians).
>
> On the other hand, DSolve gives the simple, clean answer that Integrate
> used to give:
>
> y[t]/. DSolve[{y'[t] == Sqrt[Cos[t] + 1], y[0] == 0}, y[t], t]
>
> 2*Sqrt[1 + Cos[t]]*Tan[t/2]
>
> Could it be that we need a new function such as this:
>
> RealIntegral[expr_,{x_,a_,b_}]:=
> (y[x]/. First@DSolve[{y'[x] ==expr, y[a] == 0}, y[t], t])/.x->b
>
> that would be associated with \[Integral] ? ... leaving the current
> Integrate to be associated with \[ContourIntegral]??
>
> Or perhaps a simple option for Integrate like RealLimits->True?
>
> -----
> Selwyn Hollis
> http://www.math.armstrong.edu/faculty/hollis
>
>
>
However,
Integrate[Sqrt[Cos[t] + 1], {t, 0, x},
Assumptions -> x $B":(B Reals&&-Pi<=x<=Pi]
2*Sqrt[2]*Sin[x/2]
so I can't really see what is there (in this case!) to complain about?
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/