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Integral operators

Derivative operators

# # diff

# # # # int

diff(1) from f(x) menu or functions panel

Maxima Definite Integral numerical operator from functions panel

# # # diff

diff(3) from f(x) menu only

Maple Indefinite Integral int(2) ... f(x) menu

# # int

# # n diff

n can be specified outside

11 x 2^+ / x int maple x atan

x f 11 x 2^+ / :

n 2 :

x 3^(x n diff 6 x *

Maple recognises the Maxima Int operator via the "transit integrand instruction" ............. f(u)du ...............

Note that Smath derivative operator takes "entire fnct" ONLY, unlike Mathcad Ridder deriv operator that takes piecewise fncts.

u f u 0 x int maple x atan

Copy/paste in text

BTW [By The Way, by same token ... while in context] diff(#,#),diff(#,#,n) are the Lagrange "Derivative Operator(s)".

# # n diff

# # diff

x 3^(diff x 3^diff

purely decorative in text, Germanic tyle is two dots.

It seems to me, that it is not possible to integrate an integral.[Olik] YES: you can integrate an integral in the sense of Σumming elements of the integral curve as a "cumulative integration"

An integral is an "Integral curve", i.e: the "Indefinite Integral". When you integrate an integral, you calculate the surface area under the integral curve betwenn limits [a, b], i.e: "the Definite Integral" In English literature: "AntiDerivative" synonymous of "Integral Curve".

As you open this work sheet, it calculates in ~ 10 sec. All transit plots are disabled. Enable them as you visit.

x f 1 x / :

x f x 12 int 0.693147180559949

x f x 12 int 2 ln ≡ total area under the Integral Curve within limits [1..2] ≡

x f x 13 int 1.09861228866811

This is Maxima symbolic, set the Integration accuracy > default 100

x f x 13 int maple 3 ln

3 ln 1.09861228866811

x f x int 1 x / x int

Maxima does not find the integral curve

1 x / x int maple x ln

Maple is more understanding !

add the range of the definite integral

x f x a b int surface area under f(x) between [a, b] ≡

a 1011 f 22 mat :

b 2022 f 22 mat :

x f a b 3 1 sys

as you can see from ...int(f(x),x,a,b) the definite integral of an integral curve within specified limits is a single value, no matter how the "definite integration" is done internally from the integration operator ... Romberg, Simpson, RK integrator ,,, Generally in Enginering/Scientific, we need the evolution of this integration over the running variable of integration, i.e: from a fixed low limit [a] as [b] varies. As this evolution progresses over [b] increases is called "cumulative integral" [cumulative integration]. In statistics, this cumulative integration is CDF CDF = cumulative distribution function.

..a..

..b..

x f x 0.5 3 int maple 6 ln

x f x low 3 int maple 3 low / ln

The ln(x) source generating function

x f x ln 2 1 sys

In this plot, ln(x) is simply the numerical algo from the computing machinery source code, i.e: an extended cumulative integral !

x f x ln - x 12 solve 1.7632

x f x 1 1.7632 int 0.5671

1 1 x 2 ^ + / x atan 2 1 sys x atan x diff

x F 11 x 2^+ / :

11 x 2^+ / x int maple x atan

x F x 12 int 0.321750554396643

<= On-line integration. Result is the value of the surface area [a,b] i.e: the last value of the cumulative integral.

Both q(x) are not interpretable, i.e: definitions don't match Smath internal plotting code. int(F(x),x) ... int(F(x),x,1,x)

x F x 1 x int

x F x int

x CumulIntegral x F x int maple :

C ∞ - atan : π 2 / -

Cumulative Integral of  F(x) plot x 10 - 10 9.9 - range : i 1 x rows range q i el x i el CumulIntegral C - : u i el x i el : 21 line for u q augment 31 line :

Natural complete cumulative

x CumulIntegral C - x F 0 CumulIntegral C - 3 1 sys

Cumulative Integral of F(x)

plot x F 2 1 sys

Note that the Cumulative Integral always starts @ 0, and continues in the direction of the value of the cumulated increments of the area under the integral curve, thus the need to account for the constant of integration 'C'. Note that for the discrete cumulative between the specified limits the Finite Differences is less confusing.

Advaced Calculus [Transit operators]

An important knowledge about the Smath integration operator and the Maple companionship.

u F 11 x 2^+ / :

z Int u F u x - x int :

z Int 2 x * 1 x 2^+ / ≡

z Int u F u 0 x int ≡ x 1 x 2^+ / ≡

x Int z Int :

Recast the transit operator

Transit numerical operator

x Int

On-line point value of Integral curve

1.23456789 Int 0.9782

Now, the numerical integrator looks like it has been "consumed" let say it is a "one time use". So, let redefine it and explore what more general work it does.

u f 11 x 2^+ / :

z Int u f u x - x int :

z Int 2 x * 1 x 2^+ / ≡

z Int u f u 0 x int ≡ x 1 x 2^+ / ≡

x f x int maple x atan

u f u x - x int maple 2 x * 1 x 2^+ /

u f u 0.5 x int maple 2 x * 1 - 21 x 2^+ (* /

u F u xx x int maple x xx - 1 x 2^+ /

Bidirectional Analytics of f(u)

z Int x f x f x diff 3 1 sys x 1 x 2 ^ + / 2 x * 1 x 2 ^ + / 2 x * 1 - 2 1 x 2 ^ + ( * / 3 1 sys

In my own definition, integrals that don't come easily from the formal curricular, I call them "GID Integrals" GID = Get It Done !

Resolving a GID ambiguity

In Int(z), "z" is a dummy transitional scalar that enables the plot over the canvas, a "resourcing vanishing iterator" that must be redefined in term of the variable of integration "x" for the on-line point evaluation and by same token re-enables the quick plot in its default "x scalar". In Φ(u), "u" is also a transitional iterator that interprets the numerical integration operator int(Φ(u),u,-x,x). It is not coincidence it works this way, simply that it has been properly coded GID !

u Φ 11 x 2^+ / :

z Int u Φ u x - x int :

Tools/Options/Calculation/Integral accuracy 10000 ≡

x Plot u Φ u x - x int maple : 2 x * 1 x 2^+ /

Here, don't recast the transit operator

8 - Int 0.246153846153847 -

8 Plot 0.246153846153846

x Δ x Int x Plot - :

X 4 :

sanity check

XY X X Int + 20 red 15 mat :

Very slow plot ~ 1min.20

x Int XY 2 1 sys

Very fast plot

x Plot XY 2 1 sys

Ad absurdum proof

x Δ

x Fnct 11 x 2^+ / :

Redefine for a new suite

u Fnct u x - x int maple 2 x atan *

u Fnct u 0 x int maple x atan

u Fnct u 0.5 x int maple x atan 12 / atan -

u Fnct u xx x int maple x atan xx atan -

x atan 2 x atan * x atan 1 2 / atan - 3 1 sys

Jean Giraud: 2015_12_08

x y U x atan y atan - :

y 1 :

x y U x 2^10 y * + 1 x 2^+ / :

x y U

x y U1 x 2^5 y * - 1 x 2^+ / :

U U1 2 1 sys x y U1