From 9acc79735bd9f03c04968e670f45fd48d33ecbda Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Mon, 23 Feb 2026 20:54:53 -0300 Subject: [PATCH] Discusses techniques for integration. --- .gitignore | 11 +++++++ calculus.pdf | Bin 763978 -> 768098 bytes calculus.tex | 84 +++++++++++++++++++++++++++++++++++++++++++++------ 3 files changed, 86 insertions(+), 9 deletions(-) create mode 100644 .gitignore diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..75f5da8 --- /dev/null +++ b/.gitignore @@ -0,0 +1,11 @@ +*.bib +*.bbl +*.toc +*.aux +*.fdb_latexmk +*.fls +*.log +*.out +*.run.xml +*.bcf +*.blg diff --git a/calculus.pdf b/calculus.pdf index af0377b454e7dd2a4c7b81d95062e88b17569cca..24ff719f45dadebbc22bacdb89362226c33e857d 100644 GIT binary patch delta 72931 zcmYJaV|-mv^T(T<*fttlC$?=gMq}GPVPoS2jg7{()7WmT#&(n3KF@Ri_r9AwYu3!{ z{bqkZzcp)omSyjnrA`(O47>qgHyXf2BmB>LMQ1?(?^+>YH3|@l!9wza`?~;GjfrG6 z@F~#@SdD7*e2|cQjWG%CVh%A+%a5U6dy(1w;da|GOj) 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Precisaria repensá-lo. \section{Técnicas de integração} +Sem dúvida que a técnica mais fundamental de integração é o teorema +fundamental. A partir dele, a gente estabelece regras primitivas de +integração como $\int 1/x \dx = \ln x$ uma vez que sabemos derivar +$\ln x$. Similarmente, a regra do quociente (de derivação) nos entrega +que a derivada de $\tg x$ é $\sec^2 x$ e, portanto, pelo teorema +fundamental, obtemos +% + $$\int \sec^2 x \dx = \tg x + c.$$ +% +Isso não é lá notícia tão boa porque significa que teremos que +memorizar muitas derivadas. Por exemplo, qual a derivada de $\cot x$? +Primeiro lembre-se que $\cot x = 1/\tg x$. Daí aplique a regra do +quociente e obtenha $- sec^2 x / \tg^2 x = - 1/\sin^2 x = - \csc^2 x$, +implicando que $\csc^2 x \dx = - \cot x + c$, como +desejado. Similarmente, obtém-se assim muitas outras integrais +fundamentais como $\int \sec x \tg x \dx = \sec x + c$ uma vez que a +derivada de $\sec x = \sec x \tg x$. Infelizmente, o único jeito de +fazer uso dessas regras é memorizar cada uma por prática. Matemáticos +fazem manuais de integração em que eles catalogam as importantes. A +gente não vai fazer isso, não, porque seria um trabalho muito mais +monumental, mas podemos começar a pensar em computar a derivada de +todas as funções que conhecemos como $\csc x$, $\sinh x$, $\cosh x$ +{\em et cetera}. + A regra da substituição é a regra da cadeia e a regra de integração por partes é a regra do produto. As outras regras são ou muito óbvias ou pouco úteis. Por exemplo, teria como extraírmos uma regra de @@ -4982,21 +5005,64 @@ incógnita a ser descoberta. Há integrais divertidas de se computar de onde se extrai, por exemplo, a bonita fórmula % - $$\int x^n e^x \dx = e^x \left(\sum_{i=0}^n (-1)^i \frac{n!}{n - i + 1} x^{n - i}\right) + c.$$ + $$\int x^n e^x \dx = e^x \left(\sum_{i=0}^n (-1)^i \frac{n!}{(n - i)!} x^{n - i}\right) + c$$ % +pra todo natural $n$. + Por exemplo, $\int x^5 e^x \dx = e^x (x^5 - 5 x^4 + 20 x^3 - 60x^2 + 120x - 120) + c$. Essa fórmula vem da integração por-partes. Escolhe-se sempre o termo $x^i$ como função $u$ e $e^x$ como função $v$. Aplica-se a integração por-partes $n$ vezes (recursivamente) e em seguida substitui-se de volta em direção ao -problema original. Okay, agora tragamos nossa atenção para as -integrais de funções trigonométricas. +problema original. Na verdade o que fizemos pra obter essa fórmula foi +computar uma {\em reduction formula} pra integral de $x^n e^x$. +(Veja, por exemplo, o que faz Stewart no exemplo 6 da seção 7.1 da +nona edição.) A cultura, portanto, é expressar essas fórmulas através +de uma recorrência. O que fizemos foi traduzir a recorrência pra um +laço-for iterativo, o que já é alguma vantagem porque assim a gente +consegue ver o padrão mais facilmente e nos permite memorizar certos +resultados, como a integral de $x^5 e^x$. Outros padrões talvez não +fossem tão fáceis de se obter ou memorizar, como o próprio exemplo 6 +da seção 7.1 da nona edição de Stewart. O fato, entretanto, é que uma +fórmula de redução obtida pela integração por partes só é útil se ela +for de fato uma fórmula de redução, é claro---a integral recorrente +precisa ter um grau menor que a integral que se deseja calcular. + +Uma outra observação sobre a integração por partes é que às vezes a +aplica-se ao produto artificial $f(x)\cdot 1$. Por exemplo, pra +integrar $\tg^{-1} x$, escolhe-se $u = \tg^{-1} x$ e $\dv = 1$ de forma +que $\du = 1/(1 + x^2) \dx$ e $v = x$. Daí +% + $$\int \tg^{-1} x \dx = x \tg^{-1} x - \int \frac{x}{1 + x^2} \dx,$$ +% +que pode ser então resolvida pela substituição $t = 1 + x^2$ na +integral resultante porque $(1/2x) \dt = \dx$ e aí ficamos com +$x\tg^{-1}x - \ln|1 + x^2| + c$, como desejado. + +Okay, agora tragamos nossa atenção para as integrais de funções +trigonométricas. + +\subsection{A integral de funções trigonométricas} + +Começamos por observar que integrar $\tg x$ não é difícil porque ela é +obtida por uma substituição trivial. Primeiro escreva $\tg x = \sin x +/ \cos x$. Em seguida declare $u = \cos x$ de forma que $-\du/(\sin x) += \dx$, implicando +% + $$\int \tg x \dx = -\int 1/u \du = -\ln |\cos x| + c = ln |\sec x| + c,$$ +% +como desejado. Agora vamos subir o grau. + +Feito essa observação, a gente também observa que já sabemos integrar +$\sin x$ e $\cos x$, mas que precisamos pensar em potências delas como +$\sin^2 x$, $\cos^2 x$ {\em et cetera}. Nenhma dessas duas sai por +por-partes---experimente. Obtemos $\int \sin^2 x \dx = -\sin x \cos x ++ \int \cos^2 \dx$, exigindo-nos a integral de $\cos^2 x$. Se +tentarmos por-partes a integral de $\cos^2 x$, precisaremos da +integral de $\sin^2 x$, estabelecendo-se assim um ciclo do qual não +conseguiremos sair. Agora, a sorte não é a mesma se desejarmos pelas +integrais de $\sin^3 x$ ou $\cos^3 x$. -%% \subsection{A integral de funções trigonométricas} -%% -%% Começamos por observar que integrar $\tg x$ não é difícil porque ela é -%% obtida por uma substituição trivial. Primeiro escreva $\tg x = \sin x -%% / \cos x$. \section{Studying functions}