intgrf = fnint(f,value)
intgrf = fnint(f,value) is the description of an indefinite integral of the univariate function whose description is contained in f. The integral is normalized to have the specified value at the left endpoint of the function's basic interval, with the default value being zero.
The output is of the same type as the input, i.e., they are both ppforms or both B-forms. fnint does not work for rational splines nor for functions in stform.
Indefinite integration of a multivariate function, in coordinate directions only, is available via fnder(f,dorder) with dorder having nonpositive entries.
The statement diff(fnval(fnint(f),[a b])) provides the definite integral over the interval [a .. b] of the function described by f.
If f is in ppform, or in B-form with its last knot of sufficiently high multiplicity, then, up to rounding errors, f and fnder(fnint(f)) are the same.
If f is in ppform and fa is the value of the function in f at the left end of its basic interval, then, up to rounding errors, f and fnint(fnder(f),fa) are the same, unless the function described by f has jump discontinuities.
If f contains the B-form of f, and t1 is its leftmost knot, then, up to rounding errors, fnint(fnder(f)) contains the B-form of f – f(t1). However, its leftmost knot will have lost one multiplicity (if it had multiplicity > 1 to begin with). Also, its rightmost knot will have full multiplicity even if the rightmost knot for the B-form of f in f doesn't.
Here is an illustration of this last fact. The spline in sp = spmak([0 0 1], 1) is, on its basic interval [0..1], the straight line that is 1 at 0 and 0 at 1. Now integrate its derivative: spdi = fnint(fnder(sp)). As you can check, the spline in spdi has the same basic interval, but, on that interval, it agrees with the straight line that is 0 at 0 and -1 at 1.
See the examples "Intro to B-form" and "Intro to ppform" for examples.