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% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «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")
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% (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{
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\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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