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