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% (find-angg "LATEX/2015-2-C2-VR.tex")
% (find-angg "LATEX/2015-2-C2-VR.lua")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2015-2-C2-VR.tex" :end))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2015-2-C2-VR.pdf"))
% (defun e () (interactive) (find-LATEX "2015-2-C2-VR.tex"))
% (defun u () (interactive) (find-latex-upload-links "2015-2-C2-VR"))
% (find-xpdfpage "~/LATEX/2015-2-C2-VR.pdf")
% (find-sh0 "cp -v ~/LATEX/2015-2-C2-VR.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2015-2-C2-VR.pdf /tmp/pen/")
% file:///home/edrx/LATEX/2015-2-C2-VR.pdf
% file:///tmp/2015-2-C2-VR.pdf
% file:///tmp/pen/2015-2-C2-VR.pdf
% http://angg.twu.net/LATEX/2015-2-C2-VR.pdf
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%
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\input edrxaccents.tex % (find-angg "LATEX/edrxaccents.tex")
\input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex")
\input istanbuldefs % (find-LATEX "istanbuldefs.tex")
%
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\def\ddx{\frac{d}{dx}}
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\def\arcsen{\operatorname{arcsen}}
\def\sen{\operatorname{sen}}
\def\sec{\operatorname{sec}}
\def\ln{\operatorname{ln}}
\def\subst#1{\left[\sm{#1}\right]}
\def\prims #1{∫#1\,ds}
\def\primth#1{∫#1\,dθ}
\def\ints #1#2#3{∫_{s=#1}^{s=#2}#3\,ds}
\def\intu #1#2#3{∫_{u=#1}^{u=#2}#3\,du}
\def\intx #1#2#3{∫_{x=#1}^{x=#2}#3\,dx}
\def\intth#1#2#3{∫_{θ=#1}^{θ=#2}#3\,dθ}
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\def\difu #1#2#3{\left. #3 \right|_{u=#1}^{u=#2}}
\def\difx #1#2#3{\left. #3 \right|_{x=#1}^{x=#2}}
\def\difth#1#2#3{\left. #3 \right|_{θ=#1}^{θ=#2}}
% Indefinite integrals
\def\ints #1{∫#1\,ds}
\def\intt #1{∫#1\,dt}
\def\intu #1{∫#1\,du}
\def\intx #1{∫#1\,dx}
\def\intz #1{∫#1\,dz}
\def\intth#1{∫#1\,dθ}
% Definite integrals
\def\Ints #1#2#3{∫_{s=#1}^{s=#2}#3\,ds}
\def\Intt #1#2#3{∫_{t=#1}^{t=#2}#3\,dt}
\def\Intu #1#2#3{∫_{u=#1}^{u=#2}#3\,du}
\def\Intx #1#2#3{∫_{x=#1}^{x=#2}#3\,dx}
\def\Intz #1#2#3{∫_{z=#1}^{z=#2}#3\,dz}
\def\Intth#1#2#3{∫_{θ=#1}^{θ=#2}#3\,dθ}
% Difference
\def\Difs #1#2#3{\left. #3 \right|_{s=#1}^{s=#2}}
\def\Difu #1#2#3{\left. #3 \right|_{u=#1}^{u=#2}}
\def\Difx #1#2#3{\left. #3 \right|_{x=#1}^{x=#2}}
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% ____ _ _ _
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% | | / _` | '_ \ / _ \/ __/ _` | | '_ \ / _ \
% | |__| (_| | |_) | __/ (_| (_| | | | | | (_) |
% \____\__,_|_.__/ \___|\___\__,_|_|_| |_|\___/
%
{\setlength{\parindent}{0em}
\footnotesize
\par Cálculo 2
\par PURO-UFF - 2015.2
\par VR - 28/mar/2016 - Eduardo Ochs
% \par Versão: 14/mar/2016
\par Links importantes:
\par \url{http://angg.twu.net/2015.2-C2.html} (página do curso)
\par \url{http://angg.twu.net/2015.2-C2/2015.2-C2.pdf} (quadros)
\par \url{http://angg.twu.net/LATEX/2015-2-C2-VR.pdf} (esta prova)
%\par \url{http://angg.twu.net/LATEX/2015-2-C2-material.pdf}
\par {\tt eduardoochs@gmail.com} (meu e-mail)
}
\setlength{\parindent}{0em}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\B (#1 pts){{\bf(#1 pts)}}
\bsk
\bsk
% (c2q 7 "integral de Riemann")
% (c2q 9 "integral de Riemann sem partição especificada")
% (c2q 11 "TFC")
% (c2q 13 "TFC 2")
% (c2q 15 "Substituição")
% (c2q 16 "Diferenciais")
% (c2q 22 "G(x,y) = x^2 + y^2")
% (c2q 24 "Derivada da função inversa")
% (c2q 27 "Integrando funções racionais")
% (c2q 30 "Método de Heaviside")
% (c2q 32 "Integrando funções racionais impróprias")
% (c2q 34 "Integração por partes")
% (c2q 35 "Truque do `onde'")
% (c2q 36 "Tabelas de integrais")
% (c2q 37 "Substituição trigonométrica")
% (c2q 43 "Série de Taylor")
% (c2q 45 "Plano complexo")
% (c2q 48 "Grande truque: E")
% (c2q 50 "Substituição trigonométrica")
% (c2q 51 "EDOs")
% (c2q 53 "EDOs: D")
% (c2q 55 "EDOs: sen e cos vezes exp")
% (find-es "ipython" "2015.2-C2-P1")
1) \T(Total: 6.0 pts) Calcule:
$$a) \quad \B(2.0 pts) \quad \Intx 2 3 {x^4 \ln x}$$
$$b) \quad \B(2.0 pts) \quad \intx {x^3 \sqrt{1-x^2}}$$
$$b) \quad \B(2.0 pts) \quad \intx {\frac {x^2} {x^2 - x - 6}}$$
\bsk
2) \T(Total: 2.0 pts) Seja (*) esta EDO: $f'' - f' + 6f = 0$.
a) \B(1.0 pts) Encontre as soluções básicas de $(*)$.
b) \B(1.0 pts) Encontre uma solução de $(*)$ que obedeça $f(0)=0$ e $f(1)=1$.
\bsk
3) \T(Total: 2.0 pts) Seja (**) esta EDO: $f'(x) = -2x / f(x)$.
a) \B(1.0 pts) Encontre a solução geral de (**).
b) \B(1.0 pts) Encontre uma solução de (**) que obedeça $f(0)=10$.
\newpage
Método de Heaviside:
Se $f(x) = \frac{\aa}{x-a} + \frac{\bb}{x-b} + \frac{\cc}{x-c} = \frac{p(x)}{(x-a)(x-b)(x-c)}$,
então $\lim_{x \to a} f(x)(x-a) = \aa = \frac{p(a)}{(a-b)(a-c)}$.
\bsk
Substituição:
\msk
$\begin{array}{l}
\Difx a b {g(h(x))} = \Intx a b {g'(h(x))\frac{d\,h(x)}{dx}} \\
\phantom{mmm}|\,| \\
\Difu {h(a)} {h(b)} {g(u))} = \Intu {h(a)} {h(b)} {g'(u)} \\
\end{array}
$
\msk
Fórmulas:
\msk
$\begin{array}{l}
\Intx a b {f(g(x))\frac{d\,g(x)}{dx}} \\
= \Intx a b {f(u)\frac{du}{dx}} \\
= \Intu {g(a)} {g(b)} {f(u)}
\end{array}
\qquad
\begin{array}{ll}
\intx {f(g(x))\frac{d\,g(x)}{dx}} \\
= \intx {f(u)\frac{du}{dx}} & \subst{u=g(x)} \\
= \intu {f(u)} & \subst{u=g(x)}
\end{array}
$
\bsk
Substituição inversa:
$\def\a{{h¹(α)}}
\def\b{{h¹(β)}}
\begin{array}{l}
\Difx \a \b {g(h(x))} = \Intx \a \b {g'(h(x))\frac{d\,h(x)}{dx}} \\
\phantom{mmm}|\,| \\
\Difu {h(\a)} {h(\b)} {g(u))} = \Intu {h(\a)} {h(\b)} {g'(u)} \\
\phantom{mmm}|\,| \\
\Difu α β {g(u))} = \Intu α β {g'(u)} \\
\end{array}
$
\msk
Fórmulas:
\msk
$\def\a{{g¹(α)}}
\def\b{{g¹(β)}}
\begin{array}{l}
\Intu α β {f(u)} \\
= \Intx \a \b {f(u)\frac{du}{dx}} \\
= \Intx \a \b {f(g(x))\frac{d\,g(x)}{dx}}
\end{array}
\qquad
\begin{array}{ll}
\intu {f(u)} \\
= \intx {f(u)\frac{du}{dx}} & \subst{u=g(x)\\x=g¹(u)} \\
= \intx {f(g(x))\frac{d\,g(x)}{dx}} & \subst{x=g¹(u)} \\
\end{array}
$
\bsk
Substituição trigonométrica:
\msk
$
\begin{array}{ll}
\Ints a b {F(s, \sqrt{1-s^2})} \\
= \Intth {\arcsen a} {\arcsen b} {F(\senθ, \sqrt{1-\sen^2θ}) \frac{d\senθ}{dθ}} \\
= \Intth {\arcsen a} {\arcsen b} {F(\senθ, \cosθ) \cosθ} \\
\end{array}
\qquad
\begin{array}{ll}
\ints {F(s, \sqrt{1-s^2})} \\
= \intth {F(s, \sqrt{1-s^2}) \frac{ds}{dθ}} & \subst{s=\senθ \\ θ=\arcsenθ} \\
= \intth {F(s, c) c} & \subst{s=\senθ \\ c=\cosθ \\ θ=\arcsenθ} \\
\end{array}
$
\msk
$
\begin{array}{ll}
\Intz a b {F(z, \sqrt{z^2-1})} \\
= \Intth {\arcsec a} {\arcsec b} {F(\secθ, \sqrt{\sec^2θ-1}) \frac{d\secθ}{dθ}} \\
= \Intth {\arcsec a} {\arcsec b} {F(\secθ, \tanθ) \secθ \tanθ} \\
\end{array}
\qquad
\begin{array}{ll}
\intz {F(z, \sqrt{z^2-1})} \\
= \intth {F(z, \sqrt{z^2-1}) \frac{dz}{dθ}} & \subst{z=\secθ \\ θ=\arcsec z} \\
= \intth {F(z, t) zt} & \subst{z=\secθ \\ θ=\arcsec z \\ t=\tanθ} \\
\end{array}
$
\msk
$
\begin{array}{ll}
\Intt a b {F(t, \sqrt{1+t^2})} \\
= \Intth {\arctan a} {\arctan b} {F(\tanθ, \sqrt{1+\tan^2θ}) \frac{d\tanθ}{dθ}} \\
= \Intth {\arctan a} {\arctan b} {F(\tanθ, \secθ) \sec^2θ} \\
\end{array}
\qquad
\begin{array}{ll}
\intt {F(t, \sqrt{1+t^2})} \\
= \intth {F(t, \sqrt{1+t^2}) \frac{dt}{dθ}} & \subst{t=\tanθ \\ θ=\arctan t} \\
= \intth {F(t, z) z^2} & \subst{t=\tanθ \\ θ=\arctan t \\ z=\secθ} \\
\end{array}
$
\end{document}
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