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 |
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