|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2025-1-C2-int-indefinida.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2025-1-C2-int-indefinida.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2025-1-C2-int-indefinida.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2025-1-C2-int-indefinida.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2025-1-C2-int-indefinida.pdf"))
% (defun e () (interactive) (find-LATEX "2025-1-C2-int-indefinida.tex"))
% (defun o () (interactive) (find-LATEX "2024-1-C2-int-indefinida.tex"))
% (defun u () (interactive) (find-latex-upload-links "2025-1-C2-int-indefinida"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2025-1-C2-int-indefinida.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
% (code-eec-LATEX "2025-1-C2-int-indefinida")
% (find-pdf-page "~/LATEX/2025-1-C2-int-indefinida.pdf")
% (find-sh0 "cp -v ~/LATEX/2025-1-C2-int-indefinida.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2025-1-C2-int-indefinida.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2025-1-C2-int-indefinida.pdf")
% file:///home/edrx/LATEX/2025-1-C2-int-indefinida.pdf
% file:///tmp/2025-1-C2-int-indefinida.pdf
% file:///tmp/pen/2025-1-C2-int-indefinida.pdf
% http://anggtwu.net/LATEX/2025-1-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 "2025-1-C2-int-indefinida" "2" "c2m251ii" "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-V» (to "defs-V")
% «.title» (to "title")
% «.links» (to "links")
% «.links-int-indef» (to "links-int-indef")
% «.links-leithold» (to "links-leithold")
% «.links-stewart» (to "links-stewart")
% «.links-int-partes» (to "links-int-partes")
% «.links-fusaro» (to "links-fusaro")
\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")
\usepackage{colorweb} % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-LATEX "dednat7-test1.tex")
%\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")
\input edrxgac2.tex % (find-LATEX "edrxgac2.tex")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
]{geometry}
%
\begin{document}
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\drafturl{http://anggtwu.net/LATEX/2025-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2025.1-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 "dednat7load.lua"} % (find-LATEX "dednat7load.lua")
\directlua{dednat7preamble()} % (find-angg "LUA/DednatPreamble1.lua")
\directlua{dednat7oldheads()} % (find-angg "LUA/Dednat7oldheads.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)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\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")
\def\Caurl #1{\expr{Caurl("#1")}}
\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}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt
% «defs-maxima» (to ".defs-maxima")
%L dofile "Maxima2.lua" -- (find-angg "LUA/Maxima2.lua")
\pu
% «defs-V» (to ".defs-V")
%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\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")
% (c2m251iip 1 "title")
% (c2m251iia "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2025.1}
\bsk
Aula 11: integral indefinida
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://anggtwu.net/2025.1-C2.html}
\end{center}
\newpage
% «links» (to ".links")
% (c2m251iip 2 "links")
% (c2m251iia "links")
{\bf Links}
\scalebox{0.6}{\def\colwidth{16cm}\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
% «links-leithold» (to ".links-leithold")
% (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
% «links-stewart» (to ".links-stewart")
% (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
% «links-fusaro» (to ".links-fusaro")
% (find-books "__analysis/__analysis.el" "fusaro-tlatoc" "34" "fearlessly substituting variables")
% (find-books "__analysis/__analysis.el" "fusaro-tlatoc" "178" "6 Interlude: the ordering")
\par Um livro recente da Márcia Fusaro Pinto (da UFRJ):
\par \Ca{TLATOCp45} (p.34) ...fearlessly substituting variables...
\par \Ca{TLATOCp189} (p.178) 6 Interlude: the ordering of chapters...
% (find-books "__analysis/__analysis.el" "canuto-tabacco" "302" "9.1 Primitive functions and indefinite integrals")
}\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
\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)}
\end{document}
% (find-pdfpages2-links "~/LATEX/" "2025-1-C2-int-indefinida")
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2ii"
% ee-tla: "c2m251ii"
% End: