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Re: Integrate 5.0
*To*: mathgroup at smc.vnet.net
*Subject*: [mg44251] Re: Integrate 5.0
*From*: lalu_bhatt at yahoo.com (Bhuvanesh)
*Date*: Fri, 31 Oct 2003 03:01:21 -0500 (EST)
*References*: <bnnvfj$61s$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Selwyn Hollis <sh2.7183 at misspelled.erthlink.net> 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
Hi Selwyn,
You could specify the assumption that all variables in the limits of
integration are real:
In[1]:= Integrate[Sqrt[Cos[t] + 1], {t, 0, x}, Assumptions -> x
\[Element] Reals] //InputForm
Out[1]//InputForm=
If[x <= Pi && Pi + x >= 0, 2*Sqrt[1 + Cos[x]]*Tan[x/2],
Integrate[Sqrt[1 + Cos[t]], {t, 0, x}, Assumptions ->
x \[Element] Reals && (x > Pi || Pi + x < 0)]]
In[2]:= $Version
Out[2]= 5.0 for Linux (July 10, 2003)
Cheers,
Bhuvanesh,
Wolfram Research.
-----
Disclaimer: All opinions expressed are my own and do not necessarily
reflect those of Wolfram Research.
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