Subject: bug in integrate, is anyone working on this?
From: Stavros Macrakis
Date: Sat, 2 Jun 2012 14:07:35 -0400
Rich,
Doesn't it come down to the semantics of sqrt(x^2)? If "sqrt" means
"positive square root", then this is abs(x) (Maxima's normal behavior).
But if you treat sqrt as multivalued, then you need some way to decide
whether sqrt(x^2) for negative x is x=-abs(x) or -x = abs(x). One
reasonable way to do this is as the analytic continuation of sqrt(x^2) for
positive x, which would make sqrt(x^2) -> +/- x.
So the current Maxima results are arguably correct (with the arbitrary
choice of +x rather than -x over the whole range). Mathematica's result is
clever because it includes that pesky sqrt in the result. Cf.
Mathematica's integrate(sqrt(x^2),x) => x*sqrt(x^2)/2.
What does Mathematica do for definite integrals?
-s
On Sat, Jun 2, 2012 at 1:25 PM, Richard Fateman
<fateman at eecs.berkeley.edu>wrote:
> sin(x)^2/sqrt(1-cos(x)^2);
>
> integrate(%,x,-1,1) returns 0, but the integrand is
> always non-negative.
>
> integrate(%,x) returns cos(x) which is not
> right either.
>
> Mathematica gives -cot(x)*sqrt(sin(x)^2) :)
>
> This kind of integration problem is not new.
>
>
> RJF
>
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