Simple integral wrong

Paul Cally

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Sep 1, 2000, 9:57:57 PM9/1/00

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Try integrating | cos u| from u=0 to u = Pi x. Despite the integrand
being everywhere
non-negative, Mathematica 4 gives a result which jumps DOWNWARDS by 2 at

x=1/2, 3/2, 5/2, .... I thought these simple integration errors had been
sorted out by
Wolfram years ago!

Paul Cally

--

Paul Cally

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Sep 7, 2000, 10:48:42 PM9/7/00

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Paul Cally wrote:

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,

Bill Bertram

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Sep 7, 2000, 10:47:34 PM9/7/00

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Paul Cally <ca...@kronos.maths.monash.edu.au> wrote in message
news:8opmr5$1...@smc.vnet.net...

> Try integrating | cos u| from u=0 to u = Pi x. Despite the integrand
> being everywhere
> non-negative, Mathematica 4 gives a result which jumps DOWNWARDS by 2 at
>
> x=1/2, 3/2, 5/2, ....

Are you sure of that? In my copy of Math V4 the following,

Plot[Integrate[Abs[Cos[u]], {u, 0, x Pi]}], {x, 0, 3}]

yields a graph of a monotonically increasing function!

Cheers,

Bill

Albert Retey

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Sep 10, 2000, 3:21:34 AM9/10/00

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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

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