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MathGroup Archive 2003

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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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