make the interface to integrate() like the (new consistent) interface to diff()

[jasongrout]

It would be nice if the following worked, if f was a function:

sage: integrate(f, x)

sage: # Double indefinite integral wrt x
sage: integrate(f, x, x)

sage: # limits and a double integral: x is the inner integral, y is the outer integral (note that this order is backwards from Mma...they think of nested integrals as int dx int dy function, so the first parameter is the outer integral in Mma.)
sage: integrate(f, (x, 0, 1), (y, 1, 2))
sage: integrate(f, (x, 0, y), (y, 1, 2))

sage: # Double integral, x is the inner integral, y is the outer integral
sage: integrate(f, x, y)

CC: @kcrisman @sagetrac-ktkohl

Component: calculus

Issue created by migration from https://trac.sagemath.org/ticket/2787

[jicama]

comment:1

See also #1221. It would be nice if the interface here was also consistent with the interface to plot, at least when a range is specified, since both take a function, a variable, and a range.

[jicama]

comment:2

What do you propose to do about the following case:

# Current behavior, definite integral with x and y as bounds
sage: var('x,y')
sage: integrate(x^2,x,x,y)
y^3/3 - x^3/3

# Triple Integral, twice with respect to x, once with respect to y
sage: integrate(x^2,x,x,y)
x^4*y/12

# i.e. same as
sage: (x^2).integrate(x).integrate(x).integrate(y)

[jasongrout]

comment:3

As noted in #1221, I would vote with the consensus from the devel list that the syntax integrate(f, x, a, b) be deprecated in favor of integrate(f, (x, a, b)). However, you're right that this would introduce a backwards-incompatible change.