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/ | > +--------------------------------------------------------------------------+ -- Visual Analysis GmbH Internet: www.visualanalysis.com Neumarkter Str. 87 Telefon: 089 / 431 981 0 D-81673 Muenchen Telefax: 089 / 431 981 1