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MathGroup Archive 2000

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Re: Simple integral wrong

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25107] Re: Simple integral wrong
  • From: Albert Retey <albert.retey at visualanalysis.com>
  • Date: Sun, 10 Sep 2000 03:14:30 -0400 (EDT)
  • Organization: Visual Analysis
  • References: <8opmr5$15@smc.vnet.net> <8p9k2a$1l6@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi all,

This leaves even more questions, actually I would prefer the first
"result"...

Mathematica 4.0 for Linux
Copyright 1988-1999 Wolfram Research, Inc.
 -- Motif graphics initialized -- 

In[1]:= Integrate[Abs[Cos[x]],x]

Out[1]= Integrate[Abs[Cos[x]], x]

In[2]:= Integrate[Abs[Cos[u]],{u,0,x}]

                   2
Out[2]= Sqrt[Cos[x] ] Tan[x]

In[3]:= Quit


Note that also the Differentiation goes wrong (which might be th reason
for the wrong Integration after all):

Mathematica 4.0 for Linux
Copyright 1988-1999 Wolfram Research, Inc.
 -- Motif graphics initialized -- 

In[1]:= Integrate[Abs[Cos[x]],{x,0,y}]

                   2
Out[1]= Sqrt[Cos[y] ] Tan[y]

In[2]:= Integrate[Abs[Cos[x]],{x,0,y}] // InputForm

Out[2]//InputForm= Sqrt[Cos[y]^2]*Tan[y]

In[3]:= D[Sqrt[Cos[y]^2]*Tan[y],y] // Simplify

                   2
Out[3]= Sqrt[Cos[y] ]

In[4]:= Quit

 
>  I have had a number or queries about this. Sorry I hadn't made it clearer.
> Here is
> the full story:
> 
> A couple of people told me that
> 
>             Plot[Integrate[Abs[Cos[u]], {u, 0, x  Pi]}], {x, 0, 3}]
> 
> works fine. The result is monotonic increasing as expected.
> 
> But try
> 
>             Plot[Evaluate[Integrate[Abs[Cos[u]],{u,0,Pi*x}]],{x,0,3}]
> 
> and see what happens! The evaluate forces Mathematica to do the
> integral symbolically. It was doing it numerically without the Evaluate.
> 
> Or just type
> 
>              Integrate[Abs[Cos[u]],{u,0,Pi x}]
> 
> Mathematica 4 returns
> 
>                                    2
> Out[1]= Sqrt[Cos[Pi x] ] Tan[Pi x]
> 
> (Actually, I don't think Mathematica 3 can do it at all.) This
> plots as a saw-tooth. The true solution should be
> 
> Sqrt[Cos[Pi x]^2] Tan[Pi x] + 2 Floor[x + 1/2]
> 
> Mathematica misses the step functions necessary to make the solution
> continuous.
> 
> Thanks for your interest,
> 
> Paul Cally
> 
> --
> 
> +--------------------------------------------------------------------------+
> |Assoc Prof Paul Cally            |    Ph:  +61 3 9905-4471                |
> |Dept of Mathematics & Statistics |    Fax: +61 3 9905-3867                |
> |Monash University                |    paul.cally at sci.monash.edu.au        |
> |PO Box 28M, Victoria 3800        |                                        |
> |AUSTRALIA                        | http://www.maths.monash.edu.au/~cally/ |
> +--------------------------------------------------------------------------+

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