% (find-LATEX "2024-1-C2-int-indefinida.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-1-C2-int-indefinida.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-1-C2-int-indefinida.tex" "Success!!!"))
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% (defun u () (interactive) (find-latex-upload-links "2024-1-C2-int-indefinida"))
% (defun v () (interactive) (find-2a '(e) '(d)))
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%          (code-eec-LATEX "2024-1-C2-int-indefinida")
% (find-pdf-page   "~/LATEX/2024-1-C2-int-indefinida.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C2-int-indefinida.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C2-int-indefinida.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-1-C2-int-indefinida.pdf")
%   file:///home/edrx/LATEX/2024-1-C2-int-indefinida.pdf
%               file:///tmp/2024-1-C2-int-indefinida.pdf
%           file:///tmp/pen/2024-1-C2-int-indefinida.pdf
%  http://anggtwu.net/LATEX/2024-1-C2-int-indefinida.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2024-1-C2-int-indefinida" "2" "c2m241ii" "c2ii")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.defs-caepro»	(to "defs-caepro")
% «.defs-pict2e»	(to "defs-pict2e")
% «.defs-maxima»	(to "defs-maxima")
% «.defs-rednames»	(to "defs-rednames")
% «.title»		(to "title")
% «.links»		(to "links")
% «.links-int-indef»	(to "links-int-indef")
% «.links-int-partes»	(to "links-int-partes")
%
% «.int-indefinida»	(to "int-indefinida")
%
% «.r-quociente»	(to "r-quociente")
% «.tr-igualdade»	(to "tr-igualdade")
% «.42-99»		(to "42-99")



\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
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%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx21}               % (find-LATEX "edrx21.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex            % (find-LATEX "edrx21chars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
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%
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% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

\def\drafturl{http://anggtwu.net/LATEX/2024-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2024.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2024-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2024-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% «defs-T-and-B»  (to ".defs-T-and-B")
\long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
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\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}

% «defs-caepro»  (to ".defs-caepro")
%L dofile "Caepro5.lua"              -- (find-angg "LUA/Caepro5.lua" "LaTeX")
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\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca      #1{\Cahref{#1}{#1}}

% «defs-pict2e»  (to ".defs-pict2e")
%L dofile "Piecewise2.lua"           -- (find-LATEX "Piecewise2.lua")
%L --dofile "Escadas1.lua"           -- (find-LATEX "Escadas1.lua")
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
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% «defs-maxima»  (to ".defs-maxima")
%L dofile "Maxima2.lua"              -- (find-angg "LUA/Maxima2.lua")
\pu

% «defs-rednames»  (to ".defs-rednames")
% (c2m241exsubstp 6 "defs-rednames")
% (c2m241exsubsta   "defs-rednames")
\def\redname#1{{\color{Red3}\text{#1}}}
\sa    {RC}{\redname{[RC]}}
\sa   {RCL}{\redname{[RCL]}}
\sa    {II}{\redname{[II]}}
\sa  {TFC2}{\redname{[TFC2]}}
\sa{defdif}{\redname{[defdif]}}
\sa     {4}{\redname{[4]}}
\sa     {5}{\redname{[5]}}
\sa     {6}{\redname{[6]}}
\sa     {7}{\redname{[7]}}
\sa     {8}{\redname{[8]}}
\sa    {II}{\redname{[II]}}
\sa   {IIC}{\redname{[IIC]}}
\sa    {S1}{\redname{[S${}_1$]}}

\def\bigeq#1{\Bigl(#1\Bigr)}
\def\bigeq#1{\Bigl(#1\Bigr)}



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m241iip 1 "title")
% (c2m241iia   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2024.1}

\bsk

Aula 10: integral indefinida

e integração por partes

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.1-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m241iip 2 "links")
% (c2m241iia   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{12cm}\firstcol{

% (find-LATEXgrep "grep --color=auto -nH --null -e indefi 2023*.tex")
% (c2m231macacop 7 "integral-indefinida")
% (c2m231macacoa   "integral-indefinida")
% (c2m232ipp 5 "int-indefinida")
% (c2m232ipa   "int-indefinida")

% «links-int-indef»  (to ".links-int-indef")
% (find-books "__analysis/__analysis.el" "miranda" "181" "6.1 Integral Indefinida")
% (find-books "__analysis/__analysis.el" "miranda" "207" "7. Integração definida")
\par \Ca{Miranda181} 6 Integral Indefinida
\par \Ca{Miranda182} Figura 6.1: antiderivadas de $x^2$
\par \Ca{Miranda207} 7 Integração definida
\par \Ca{Miranda212} 7.2 Integral definida

\ssk

% (find-books "__analysis/__analysis.el" "leithold" "285" "5. Integração e integral definida")
% (find-books "__analysis/__analysis.el" "leithold" "286" "5.1. Antidiferenciação")
% (find-books "__analysis/__analysis.el" "leithold" "287" "5.1.3. Teorema: ...em um intervalo I")
% (find-books "__analysis/__analysis.el" "leithold" "324" "5.5. A integral definida")
\par \Ca{Leit5p2}  (p.285) 5 Integração e integral definida
\par \Ca{Leit5p3}  (p.286) 5.1 Antidiferenciação
\par \Ca{Leit5p4}  (p.287) 5.1.3 Teorema: ...em um intervalo $I$
\par \Ca{Leit5p41} (p.324) 5.5 A integral definida

\ssk

% (find-books "__analysis/__analysis.el" "stewart-pt" "337" "5.2 A Integral Definida")
% (find-books "__analysis/__analysis.el" "stewart-pt" "360" "5.4 Integrais Indefinidas")
% (find-books "__analysis/__analysis.el" "stewart-pt" "361" "primitiva mais geral sobre um dado int")
\par \Ca{StewPtCap5p16} (p.337) 5.2 A Integral Definida
\par \Ca{StewPtCap5p39} (p.360) 5.4 Integrais Indefinidas
\par \Ca{StewPtCap5p40} (p.361) primitiva geral da função $f(x)=1/x^2$

\bsk

% «links-int-partes»  (to ".links-int-partes")
% (find-books "__analysis/__analysis.el" "miranda" "199" "6.3 Integração por Partes")
% (find-books "__analysis/__analysis.el" "leithold" "531" "9.1. Integração por partes")
% (find-books "__analysis/__analysis.el" "stewart-pt" "420" "7.1 Integração por Partes")
\par \Ca{Miranda199} 6.3 Integração por Partes
\par \Ca{Leit9p4} (p.531) 9.1. Integração por partes
\par \Ca{StewPtCap7p5} (p.420) 7.1 Integração por Partes

\bsk

Um livro recente da Márcia Fusaro Pinto (da UFRJ):

% (find-books "__analysis/__analysis.el" "fusaro-tlatoc" "34" "fearlessly substituting variables")
% (find-books "__analysis/__analysis.el" "fusaro-tlatoc" "178" "6 Interlude: the ordering")
\par \Ca{TLATOCp45} (p.34) ...fearlessly substituting variables...
\par \Ca{TLATOCp189} (p.178) 6 Interlude: the ordering of chapters...


}\anothercol{



}}

\newpage


% «int-indefinida»  (to ".int-indefinida")
% (c2m241iip 3 "int-indefinida")
% (c2m241iia   "int-indefinida")
% (c2m232ipp 5 "int-indefinida")
% (c2m232ipa   "int-indefinida")


{\bf Integral indefinida}

\scalebox{0.55}{\def\colwidth{10cm}\firstcol{

Tanto o Leithold quanto o Miranda explicam a {\sl integral indefinida}
antes da {\sl integral definida}. Dê uma olhada na página de links.

\msk

{\sl Todos os modos fáceis de atribuir um significado intuitivo para
  expressões como esta aqui}
%
$$\intx{f(x)}$$

{\sl são gambiarras que funcionam mal.}

\msk

Eu vou usar esta definição aqui,

\ssk

\Ca{2fT23} (p.4) Outra definição para a integral indefinida

\ssk

e aqui tem um caso em que a definição usual quebra:

\ssk

\Ca{2fT24} (p.5) Meme: expanding brain, versão ln

\msk

}\anothercol{

Nós vamos começar usando a integral indefinida como o macaco que faz
contas sem ter idéia do significado do que está fazendo, e só depois
que tivermos bastante prática nós vamos discutir os vários jeitos de
atribuir significados intuitivos para % $\intx{f(x)}$.

\msk

A regra básica vai ser esta aqui:

$$\ga{II} = \left( \intx{f'(x)} = f(x) \right)$$

\bsk

{\bf Exercícios}

\msk

Calcule:

\ssk

%a) $\ga{II} \CME{.[f(x) := x+42 ;; f'(x) := 1]}$

%b) $\ga{II} \CME{.[f(x) := {1//2} mul x^2 ;; f'(x) := x]}$

\msk

c) Resolva os exercícios 1 a 10 daqui por chutar e testar:

\Ca{Miranda185} Exercícios 6.1

\msk

d) Entenda tudo que esta nesta página:

\Ca{Leit5p6} (p.289) 5.1.8. Teorema

}}


\newpage

% «r-quociente»  (to ".r-quociente")
% (c2m241iip 4 "r-quociente")
% (c2m241iia   "r-quociente")

{\bf A regra do quociente}

\def\x{}    \def\CD{·}
\def\x{(x)} \def\CD{}
\def\P#1{\left(#1\right)}
\def\L{\\[-11pt]}

\sa{quotient rule}{
  \begin{array}{rcl}
  \D \ddx g\x^k           &=& \D kg\x^{k-1} g'\x \\\L
  \D \ddx g\x^{-1}        &=& \D -g\x^{ -2} g'\x \\\L
  \D \ddx \frac{1}  {g\x} &=& \D -\frac{g'\x}{g\x^2} \\\L
  \D \ddx \frac{f\x}{g\x} &=& \D \P{\ddx f\x}\frac{1}{g\x} + f\x\CD\P{\ddx \frac{1}{g\x}} \\\L
  \D                      &=& \D f'\x \frac{1}{g\x} + f\x\CD\P{-\frac{g'\x}{g\x ^2}} \\\L
  \D                      &=& \D \frac{f'\x g\x  - f\x g'\x}{g\x^2} \\\L
  \D \ddx \frac{f\x}{g\x} &=& \D \frac{f'\x g\x  - f\x g'\x}{g\x^2} \\\L
  \end{array}
  }

\scalebox{0.42}{\def\colwidth{11cm}\firstcol{
\vspace*{-0.5cm}
$$
  \def\x{}    \def\CD{·}
  \def\x{(x)} \def\CD{}
  \ga{quotient rule}
$$
\bsk
$$
  \def\x{(x)} \def\CD{}
  \def\x{}    \def\CD{·}
  \ga{quotient rule}
$$

}\anothercol{
}}


\newpage

% «tr-igualdade»  (to ".tr-igualdade")
% (c2m241iip 5 "tr-igualdade")
% (c2m241iia   "tr-igualdade")

{\bf A transitividade da igualdade}

\sa{abcdefgh}{\pmat{a+b &=& c+d \\ &=& e+f \\ &=& g+h \\ a+b &=& g+h}}
\sa{29384756}{\pmat{2+9 &=& 3+8 \\ &=& 4+7 \\ &=& 5+6 \\ 2+9 &=& 5+6}}
\sa{a:=2 etc}{\bsm {a:=2 \\ b:=9 \\ c:=3 \\ d:=8 \\ e:=4 \\ f:=7 \\ g:=5 \\ h:=6}}

$$\ga{abcdefgh}\ga{a:=2 etc} = \ga{29384756}
$$


\newpage

\def\INTX#1{\intx{#1}}
\def\INTX#1{\Intx{a}{b}{#1}}
\def\x{}    \def\CD{·} \def\INTX#1{\intx{#1}}
\def\x{(x)} \def\CD{}  \def\INTX#1{\Intx{a}{b}{#1}}

\sa{int kf = k int f}{
  \begin{array}{rcl}
  \D  \INTX{f\x} &=& \D  F\x \\\L
  \D k\INTX{f\x} &=& \D kF\x \\\L
  \D \INTX{kf\x} &=& \D kF\x \\\L
  \D \INTX{kf\x} &=& \D k\INTX{f\x} \\ 
  \end{array}
  }

\sa{int f+g = int f + int g}{
  \begin{array}{rcl}
  \D \INTX{f\x}     &=& \D F\x \\ 
  \D \INTX{g\x}     &=& \D G\x \\ 
  \D \INTX{f\x}+\INTX{g\x} &=& \D F\x+G\x \\ 
  \D \INTX{f\x +      g\x} &=& \D F\x+G\x \\ 
  \D \INTX{f\x+g\x} &=& \D \INTX{f\x} + \INTX{g\x} \\ 
  \end{array}
  }

{\bf Linearidade da integral}

\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

\def\x{}    \def\CD{·} \def\INTX#1{\intx{#1}}       \def\D{}
\def\x{(x)} \def\CD{}  \def\INTX#1{\Intx{a}{b}{#1}} \def\D{\displaystyle}

$$\ga{int kf = k int f}
$$

$$\ga{int f+g = int f + int g}
$$


}\anothercol{

\def\x{(x)} \def\CD{}  \def\INTX#1{\Intx{a}{b}{#1}} \def\D{\displaystyle}
\def\x{}    \def\CD{·} \def\INTX#1{\intx{#1}}       \def\D{}

$$\ga{int kf = k int f}
$$

$$\ga{int f+g = int f + int g}
$$

}}


\newpage

% «42-99»  (to ".42-99")
% (c2m241iip 7 "42-99")
% (c2m241iia   "42-99")

\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

$$\begin{array}{rcl}
      \ga{II} &=& \bigeq{ \intx      {F'(x)} = F(x) } \\
     \ga{IIC} &=& \bigeq{ \intx      {F'(x)} = F(x) + C } \\
    \ga{TFC2} &=& \bigeq{ \Intx{a}{b}{F'(x)} = \Difx{a}{b}{F(x)} } \\
  \ga{defdif} &=& \bigeq{ \Difx{a}{b}{F(x)} = F(b)-F(a) } \\
  \end{array}
$$

\def\INTX#1{\Intx{2}{3}{#1}}  \def\DIFX#1{\Difx{2}{3}{#1}}
\def\INTX#1{\intx{#1}}        \def\DIFX#1{#1}

$$\begin{array}{rcl}
  \D \INTX{0} &=& \D \DIFX{42} \\
  \D \INTX{0} &=& \D \DIFX{99} \\
  \D \DIFX{42} &=& \D \DIFX{99} \\
  \end{array}
$$

\def\INTX#1{\Intx{2}{3}{#1}}  \def\DIFX#1{\Difx{2}{3}{#1}}
\def\INTX#1{\intx{#1}}        \def\DIFX#1{#1}
\def\INTX#1{\intx{#1}}        \def\DIFX#1{#1+C}

$$\begin{array}{rcl}
  \D \INTX{0} &=& \D \DIFX{42} \\
  \D \INTX{0} &=& \D \DIFX{99} \\
  \D \DIFX{42} &=& \D \DIFX{99} \\
  \end{array}
$$

\def\INTX#1{\intx{#1}}        \def\DIFX#1{#1}
\def\INTX#1{\Intx{2}{3}{#1}}  \def\DIFX#1{\Difx{2}{3}{#1}}

$$\begin{array}{rcl}
  \D \INTX{0} &=& \D \DIFX{42} \\
  \D \INTX{0} &=& \D \DIFX{99} \\
  \D \DIFX{42} &=& \D \DIFX{99} \\
  \end{array}
$$

}\anothercol{
}}


\newpage


\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

\def\INTX#1{\Intx{a}{b}{#1}}  \def\DIFX#1{\Difx{a}{b}{#1}}  \def\x{(x)}
\def\INTX#1{\intx{#1}}        \def\DIFX#1{#1}               \def\x{}

$$\begin{array}{rcl}
  \D \INTX{f'\x g\x + f\x g'\x} &=& \D \DIFX{f\x g\x} \\
  \D \INTX{f'\x g\x + f\x g'\x} &=& \D \INTX{f'\x g\x} + \INTX{f\x g'\x} \\
  \D \INTX{f'\x g\x} + \INTX{f\x g'\x} &=& \D \DIFX{f\x g\x} \\
  \D \INTX{f\x g'\x} &=& \D \DIFX{f\x g\x} - \INTX{f'\x g\x} \\
  \end{array}
$$

\def\INTX#1{\intx{#1}}        \def\DIFX#1{#1}               \def\x{}
\def\INTX#1{\Intx{a}{b}{#1}}  \def\DIFX#1{\Difx{a}{b}{#1}}  \def\x{(x)}

$$\begin{array}{rcl}
  \D \INTX{f'\x g\x + f\x g'\x} &=& \D \DIFX{f\x g\x} \\
  \D \INTX{f'\x g\x + f\x g'\x} &=& \D \INTX{f'\x g\x} + \INTX{f\x g'\x} \\
  \D \INTX{f'\x g\x} + \INTX{f\x g'\x} &=& \D \DIFX{f\x g\x} \\
  \D \INTX{f\x g'\x} &=& \D \DIFX{f\x g\x} - \INTX{f'\x g\x} \\
  \end{array}
$$

}\anothercol{
}}


\newpage

% «int-partes-exemplo»  (to ".int-partes-exemplo")
% (c2m232ipp 4 "int-partes-exemplo")
% (c2m232ipa   "int-partes-exemplo")


{\bf Integração por partes: um exemplo}

\def\por#1{\text{(por #1)}}
\def\por#1{\text{por #1}}

\scalebox{0.55}{\def\colwidth{7cm}\firstcol{

    Lembre que o Mathologer diz no vídeo dele que o melhor modo da
    gente aprender Cálculo é começar escrevendo idéias que a gente
    acha que devem ser verdade, e depois a gente vê se elas dão
    resultados certos e se elas fazem sentido... e se fizerem sentido
    a gente tenta formalizar elas.

    \msk

    Ele também diz -- a partir daqui, na ``lombada número 1'',

    \ssk

    \Ca{CalcEasy20:27}

    \ssk

    que a integral é a inversa da derivada, mas que $\intx{\cos x}$
    pode retornar tanto $\sen x$ quanto $42+\sen x$. As contas à
    direita são bem improvisadas, mas como eu indiquei em cima que
    elas são só uma idéia que pode estar cheia de erros o ``colega que
    seja menos meu amigo'' não vai poder reagir deste jeito aqui...

    \ssk

    \Ca{2gT20}

    \bsk

    {\bf Exercício 0:}

    Calcule $\ddx(x^2e^x - 2xe^x + 2e^x)$.

%  (eepitch-maxima)
%  (eepitch-kill)
%  (eepitch-maxima)
% f : x^2*exp(x) - 2*x*exp(x) + 2*exp(x);
% diff(f, x);

}\anothercol{

Idéia (que pode estar cheia de erros):

\bsk

$\begin{array}[t]{rcll}
  (gh)'           &\eqnp{1}& g'h + gh'                   & \por{$\ga{[DProd]}$} \\
  \intx{(gh)'}    &\eqnp{2}& \intx{g'h + gh'}            \\
  gh              &\eqnp{3}& \intx{g'h + gh'}            \\
                  &\eqnp{4}& \intx{g'h} + \intx{gh'}     & \por{$\ga{[IISoma]}$} \\
  gh \phantom{mmmmmi} &\eqnp{5}& \intx{g'h} + \intx{gh'} & \por{3 e 4} \\
  gh - \intx{g'h} &\eqnp{6}& \phantom{mmmmm} \intx{gh'}  & \por{5} \\
  \\[-5pt]
  \intx{gh'}      &\eqnp{7}& gh - \intx{g'h}             & \por{6} \\
  \\[-5pt]
  \intx{xe^x}     &\eqnp{8}& xe^x - \intx{1·e^x}         & \por{7 com $\bsm{g:=x \\ h:=e^x}$} \\
                  &\eqnp{9}& xe^x - \intx{e^x}           \\
                  &\eqnp{10}& xe^x - e^x                 & \por{$(e^x)'=e^x$} \\
  \intx{xe^x}     &\eqnp{11}& xe^x - e^x                 & \por{8, 9 e 10} \\
  \\[-5pt]
  \intx{x^2e^x}   &\eqnp{12}& x^2e^x - \intx{2xe^x}      & \por{7 com $\bsm{g:=x^2 \\ h:=e^x}$} \\
                  &\eqnp{13}& x^2e^x - 2\intx{xe^x}      & \por{$\ga{[IIMC]}$} \\
                  &\eqnp{14}& x^2e^x - 2\P{xe^x - e^x}   & \por{11} \\
                  &\eqnp{15}& x^2e^x - 2xe^x + 2e^x      \\
  \end{array}
$

}}




\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}




[RC] =
[RProd] =
[RMC] =
[defdif] =



\intx{xe^x} = xe^x - \intx{1·e^x}
            = xe^x - \intx{e^x}
            = xe^x - e^x
           

[defdif] [F(x):=42 \\ a:=2 \\ b:=7] = \P{\Difx{2}{3}{42} = 42-42}


Troque por H e K

\intx{(G(x)+H(x))'} = G(x)+H(x)
\intx{G'(x)+H'(x) } = G(x)+H(x)
\intx{G'(x)}        = G(x)
\intx{H'(x)}        = H(x)
\intx{G'(x)+H'(x) } = G(x)+H(x)
\intx{G'(x)+H'(x) } = \int{G'(x)}+\int{H'(x)}
\intx{f(x)+g(x)}    = \int{f(x)} +\int{g(x)}


\int{kH'(x)} = kH(x)
\int {H'(x)} =  H(x)
k\int{H'(x)} = kH(x)
\int{kH'(x)} = k\int{H'(x)}
\int{kf(x)}  = k\int{f(x)}









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