kill(all)$

Maxima : math :use antidiff = integration , version 1.0
peter.vlasschaert@gmail.com,05/08/2021

differential notation:
===============

f: diff(f(x), x);
'at(f, [x=p]);

→  definition

f(x):=x^3;

→ @ x=p ,differentation

'at(diff(f(x),x),x=p);

→  evaluation 

ev(%,nouns);

load packages
============

load(antid);

when differential: diff(g(x),x) under the integral sign
======================================
first        =  x
second =  g(x)

→ function to integrate:sol

f1:g(x)*exp(g(x))*diff(g(x),x);

sol:integrate(f1,x);

sol1:antidiff(f1,x,g(x));

sol11:sol =sol1;

→ list structure

a1:antid (f1, x, g(x));

sol2:diff(sol1,x);

sol3:ratsimp(sol2);

representation:
===========

rp1:sol = a1[1]+constant1;

when differential: diff(x,x) =1 under the integral sign
=======================================
first         = x
second  = x

→ function to integrate:sola

f2:x^2;

sola:integrate(f2,x);

How to use : antidiff(function,first,second) = integrate(f2,x)

sol1a:antidiff(f2,x,x);

→ list structure

a2:antid (f2,x,x);

find :f2

sol2a:diff(sol1a,x);

representation: integration 'see above relation between antidiff'
=================================================

rp2:'integrate(f2,x) =integrate(f2,x)+constant2;

second way with for integration  : antidiff

proof : sol
=====

first step : integration  exp(g(x))
-------------------------------------------

diff(u*v);

example :
=======

→ math :

p1:'integrate(exp(2*x),x);

p1a:changevar(p1, u=2*x, u, x);

p1b:ev(p1a,nouns);

p1c:subst([u=2*x],p1b);

→ maxima:

p1d:'integrate(exp(2*x),x)=integrate(exp(2*x),x);

general
======

pp:'integrate(exp(g(x))*diff(g(x), x), x)= integrate(exp(g(x))*diff(g(x), x), x);

second step : integration  (%e^g(x)*g(x)*('diff(g(x),x,1)))
---------------------------------------------------------------------------

(p)artial (i)ntegration : PI
----------------------------------
math :
-------------------------------------------------
u  →  du  (differentation)
dv →  v    (integration)
-------------------------------------------------
integrate(u*dv) =u*v -integrate(v*du)
-------------------------------------------------

diff(u*v);

p2a:u=g(x);

p2b:diff(p2a);

→ del(u) =du

p2b1:subst([del(u)=du],lhs(p2b))=subst([del(x)=dx],rhs(p2b));

maxima : can't use this 'only representation',problem :dx

p2b2:du=coeff(rhs(p2b1),dx);

rem :working around : to solve problem in Maxima

p2d:v=rhs(pp);

partial integration: u*v -integrate(v*du,x)
------------------------------------------------------

1e) v*du

p2ee:subst([p2b2,p2d],v*du);

2e) PI

p2e:sol=subst([p2a,p2d],u*v-integrate(p2ee,x));

rem,:integrate(v*du)=lhs(pp)

p2e1:lhs(p2e)=facsum(ev(rhs(p2e),integrate));

check:rhs(sol11)  ? rhs(p2e1)

is(rhs(sol11)=rhs(p2e1));

maxima : 'use packages :load(antid);without use PI
maple    : doesn't need packages.(default)

appendix:
=======

--------------------------------------------------------------------------------------------------

examples : Maxima manual
=====================

(%i1) load ("antid")$
(%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                            z(x)  d
(%o2)                y(x) %e     (-- (z(x)))
                                  dx
(%i3) a1: antid (expr, x, z(x));
                       z(x)      z(x)  d
(%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                       dx
(%i4) a2: antidiff (expr, x, z(x));
                            /
                     z(x)   [   z(x)  d
(%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                            ]         dx
                            /
(%i5) a2 - (first (a1) + 'integrate (second (a1), x));
(%o5)                           0
(%i6) antid (expr, x, y(x));
                             z(x)  d
(%o6)             [0, y(x) %e     (-- (z(x)))]
                                   dx
(%i7) antidiff (expr, x, y(x));
                  /
                  [        z(x)  d
(%o7)             I y(x) %e     (-- (z(x))) dx
                  ]              dx
                  /

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