> sage: var('x,y')
> (x, y)
> sage: f = -x-y
> sage: integrate(f)
> -1/2*x^2 - x*y
> sage: integrate(f+x)  # unambiguous?
> -1/2*y^2
> sage: integrate(f+y)  # unambiguous?
> -1/2*x^2
> sage: integrate(f) + integrate(x)
> -x*y
> sage: integrate(f) + integrate(y)
> -1/2*x^2 - x*y + 1/2*y^2
>

I meant single-variable.  No, none of these are unambiguous.

>
>
> >>> While (1) and (2) syntaxes are encouraged, (3) will
> >>> remain valid until we sort out the coersion issue
> >>> and update all doctests, tutorial etc. BTW, I did update
> >>> some of the doctests including the docstrings that you get
> >>> via "integrate?"
>
> >> Sounds like we should throw a deprecation warning on it.
>
> > Yes, this would definitely require it.  There should conceivably be a
> > check for whether there are x, a, and b, and if a and b are both
> > numeric types, allowing (3) indefinitely in that case (as opposed to
> > integrate(f,x,a,b) where a and b are symbolic endpoints).
>
> I'm -1 to having a different meaning based on whether or not
> something is "numeric." For example,
>
> integrate(f, x, a, b).subs(a=1, b=2) != integrate(f, x, 1, 2)
>
> bothers me.
>

Hmm, I didn't think about that.  (Since I don't use subs much.)

> > On the plus side, FJ is correct that it is impossible to have multiple
> > integration in an unambiguous way without removing (3) eventually (so
> > as to allow integrate(f,x,y,x) and the like).  My preference would be
> > to require instead integrate(f,(x,),(y,),(x,)) or integrate(f,[x],[y],
> > [z]), but I think that those would not prove popular.
>
> Yes, I think we should support this as well, at least the tuple version.
>
> - Robert
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