<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

C\303\241lculo II

Li\303\247\303\243o 11: Integra\303\247\303\243o de Fun\303\247\303\265es Racionais

Sabe-se, do Teorema Fundamental do C\303\241lculo, que toda fun\303\247\303\243o cont\303\255nua possui uma anti-derivada. Especificamente, se f \303\251 cont\303\255nua no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=, ent\303\243o 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 \303\251 diferenci\303\241vel no intervalo aberto (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMg==), e 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 neste intervalo. Portanto, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== \303\251 uma anti-derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== .

Por outro lado, nem toda fun\303\247\303\243o (cont\303\255nua) possui uma anti-derivada elementar, que pode ser definida simplesmente como uma anti-derivada formada a partir das fun\303\247\303\265es estudadas no primeiro curso de C\303\241lculo. J\303\241 estudamos alguns exemplos.

int(exp(-x^2), x);

LCQqJkkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiIy1JJGVyZkc2JEYlSShfc3lzbGliRzYiNiNJInhHRi1GJ0Ym

int(sqrt(x^3 + 1), x);

LCYqJkkieEc2IiIiIiwmKiRGJCIiJEYmRiZGJiNGJiIiIyNGKyIiJiouLCYjRilGK0YmKiZeIyMhIiJGK0YmRilGKkYmRiYqJiwmRiRGJkYmRiZGJkYvRjRGKiomLChGJEYmRjNGJkYxRiZGJiwmIyEiJEYrRiZGMUYmRjRGKiomLChGJEYmRjNGJiomXiNGKkYmRilGKkYmRiYsJkY6RiZGPkYmRjRGKkYnRjMtSSpFbGxpcHRpY0ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JCokRjVGKiokKiZGQEYmRjlGNEYqRiYjIiInRi0=

As fun\303\247\303\265es erf e EllipticF que aparecem nestas anti-derivadas n\303\243o s\303\243o familiares: n\303\243o se encontram no livro de Stewart e, portanto, n\303\243o s\303\243o consideradas elementares. No entanto, \303\251 interessante conhecer quais fun\303\247\303\265es possuem anti-derivadas elementares. Nesta se\303\247\303\243o, veremos que toda fun\303\247\303\243o racional possui anti-derivada. Para isto, mostraremos explicitamente como tais fun\303\247\303\265es podem ser anti-diferenciadas. (Lembremo-nos de que uma fun\303\247\303\243o racional possui a forma 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, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== s\303\243o polin\303\264mios.) A id\303\251ia b\303\241sica consiste em mostrar, inicialmente, que qualquer fun\303\247\303\243o racional pode ser escrita como uma soma de certos tipos especiais de fun\303\247\303\265es racionais. Em seguida, as anti-derivadas para cada uma destas fun\303\247\303\265es deve ser calculada. Estes tipos especiais de fun\303\247\303\265es s\303\243o chamadas fra\303\247\303\265es parciais. Come\303\247aremos por estudar alguns tipos de fra\303\247\303\265es parciais que ser\303\243o necess\303\241rias.

<Text-field style="Heading 1" layout="Heading 1">Fra\303\247\303\265es Parciais</Text-field>

Consideremos a express\303\243o racional abaixo,

f := (x^7 + 6*x^5 - 3*x^2 + 7) / (2*x^5 + 21*x^4 + 44*x^3 - 24*x^2 + 144*x - 432);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoKiYsKiokSSJ4R0YoIiIoIiIiKiRGMSIiJiIiJyokRjEiIiMhIiRGMkYzRjMsLkY0RjgqJEYxIiIlIiNAKiRGMSIiJCIjV0Y3ISNDRjEiJFciISRLJUYzISIiNyNGLg==

Como deseja-se expressar separadamente o numerador e o denominador desta express\303\243o, deve-se utilizar os comandos numer e denom:

P := numer(f);

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Q := denom(f);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJRR0YoLC4qJEkieEdGKCIiJiIiIyokRjAiIiUiI0AqJEYwIiIkIiNXKiRGMEYyISNDRjAiJFciISRLJSIiIjcjRi4=

(Da mesma forma, pode-se considerar inicialmente P e Q , definindo ent\303\243o f como sendo seu quociente.)

Neste exemplo, o grau do numerador de f \303\251 maior ou igual ao grau do denominador. Sendo assim, \303\251 poss\303\255vel dividir P por Q, de forma a obter-se um quociente q e um resto r, tal que 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 , onde o grau de r \303\251 menor que o grau de Q. A express\303\243o f pode ser reescrita nesta forma, a qual denotaremos por f1.

q := quo(P, Q, x);

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r := rem(P, Q, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoLCwqJEkieEdGKCIiJSMhJXRmIiIpKiRGMCIiJCMhJVZYIiIjKiRGMEY5IiUrQEYwISVhISoiJmwuIyIiIjcjRi4=

f1 := q + r/Q;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmMUdGKCwqKiRJInhHRigiIiMjIiIiRjFGMCMhI0AiIiUjIiR4JCIiKUYzKiYsLCokRjBGNiMhJXRmRjkqJEYwIiIkIyElVlhGMUYvIiUrQEYwISVhISoiJmwuI0YzRjMsLiokRjAiIiZGMUY8IiNARj8iI1dGLyEjQ0YwIiRXIiEkSyVGMyEiIkYzNyNGLg==

Pode-se facilmente verificar que a express\303\243o f1 \303\251 igual a f:

simplify(f1);

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Como o polin\303\264mio q pode ser facilmente anti-diferenciado, o problema resultante consiste em resolver a anti-diferencia\303\247\303\243o da fun\303\247\303\243o racional pr\303\263pria LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYxNiVRIlFGJ0Y0RjcvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkQvJSliZXZlbGxlZEdRJmZhbHNlRic= . Chamaremos esta express\303\243o de f2:

f2 := r/Q;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmMkdGKComLCwqJEkieEdGKCIiJSMhJXRmIiIpKiRGMSIiJCMhJVZYIiIjKiRGMUY6IiUrQEYxISVhISoiJmwuIyIiIkY/LC4qJEYxIiImRjpGMCIjQEY2IiNXRjshI0NGMSIkVyIhJEslRj8hIiI3I0Yu

Em princ\303\255pio, o passo a seguir \303\251 sempre poss\303\255vel, embora seja imposs\303\255vel de ser realizado na pr\303\241tica: fatora-se o denominador em fatores quadr\303\241ticos lineares e irredut\303\255veis.

factor(Q);

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Feita a fatora\303\247\303\243o, pode-se escrever f2 como uma soma de fra\303\247\303\265es parciais:

f3 := convert(f2, parfrac, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmM0dGKCwqKiQsJkkieEdGKCIiIiIiJ0YyISIjIyInJHBFJCIkKycqJiwmRjEhJXA7ISM7RjJGMiwmKiRGMSIiI0YyIiIlRjIhIiIjRjIiJisrIiokLCZGMUY+ISIkRjJGQCMiJV4hKSImK10lKiRGMEZAIyEoSiM9biImKyE9NyNGLg==

Deve estar claro que, se for poss\303\255vel anti-diferenciair cada termo desta soma, ent\303\243o a express\303\243o f original tamb\303\251m poder\303\241 ser anti-diferenciada. Nas se\303\247\303\265es seguintes, estudaremos como anti-diferenciar qualquer tipo de fra\303\247\303\243o parcial. Por\303\251m, apresentaremos inicialmente alguns tipos de fra\303\247\303\265es parciais que podem surgir, e como estas relacionam-se com a fatora\303\247\303\243o do denominador.

Duas observa\303\247\303\265es:

1. O comando convert pode realizar a divis\303\243o polinomial inicial, assim como a decomposi\303\247\303\243o subsequente. Utilizando a fun\303\247\303\243o original f, por exemplo, tem-se

convert(f, parfrac, x);

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2. A fatora\303\247\303\243o \303\251 a maior dificuldade em todo o processo de fra\303\247\303\265es parciais: todos os outros passos s\303\243o autom\303\241ticos (embora o procedimento como um todo seja extremamente longo e tedioso). Por\303\251m, n\303\243o existe um m\303\251todo sistem\303\241tico para efetuar a fatora\303\247\303\243o de um polin\303\264mio. H\303\241 apenas um teorema que garante que a fatora\303\247\303\243o existe.

<Text-field style="Heading 2" layout="Heading 2">Quest\303\243o 1</Text-field>

Apresentamos a seguir algumas express\303\265es racionais. Deve-se utilizar o comando convert para reescrev\303\252-las como fra\303\247\303\265es parciais, e observar os tipos de termos que podem surgir. Deve-se ent\303\243o buscar uma regra que relacione os tipos de fra\303\247\303\265es parciais que ocorrem em determinada decomposi\303\247\303\243o com a fatora\303\247\303\243o do numerador. Em seguida, pode-se experimentar tal regra com outros exemplos e verificar se \303\251 poss\303\255vel prever os tipos de fra\303\247\303\265es parciais:

Solu\303\247\303\265es.

g1 := 1/(x^2 - 9);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnMUdGKCokLCYqJEkieEdGKCIiIyIiIiEiKkYzISIiNyNGLg==

convert(g1,parfrac,x);

LCYqJCwmSSJ4RzYiIiIiIiIkRichIiIjRikiIicqJCwmRiVGJyEiJEYnRikjRidGKw==

g2 := (3*x + 5)/(x^2 - 9);

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convert(g2,parfrac,x);

LCYqJCwmSSJ4RzYiIiIiIiIkRichIiIjIiIjRigqJCwmRiVGJyEiJEYnRikjIiIoRig=

g3 := (3*x + 5)/(x^2 - 9)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnM0dGKComLCZJInhHRigiIiQiIiYiIiJGMywmKiRGMCIiI0YzISIqRjMhIiM3I0Yu

convert(g3,parfrac,x);

LCoqJCwmSSJ4RzYiIiIiIiIkRichIiMjISIiIiIqKiQsJkYlRichIiRGJ0YpIyIiKCIjPSokRiRGKyMiIiYiJDMiKiRGLkYrIyEiJkY2

g4 := (2*x^3 + x)/(x^2 - 9)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnNEdGKComLCYqJEkieEdGKCIiJCIiI0YxIiIiRjQsJiokRjFGM0Y0ISIqRjQhIiM3I0Yu

convert(g4,parfrac,x);

LCoqJCwmSSJ4RzYiIiIiIiIkRichIiMjISM+IiM3KiQsJkYlRichIiRGJ0YpIyIjPkYsKiRGJCEiIkYnKiRGLkYzRic=

g5 := (x^3 - 2*x^2 + 3)/(x^2 - 9)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnNUdGKComLCgqJEkieEdGKCIiJCIiIiokRjEiIiMhIiNGMkYzRjMsJkY0RjMhIipGMyEiJDcjRi4=

convert(g5,parfrac,x);

LC4qJCwmSSJ4RzYiIiIiIiIkRichIiQjIiIoIiNPKiQsJkYlRidGKUYnRikjRiciIz0qJEYkISIjIyEiIiIjNyokRi5GMiNGJyIjQyokRiRGNCNGNCIkVyIqJEYuRjQjRidGOw==

g6 := x/(x^2 + 9);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnNkdGKComSSJ4R0YoIiIiLCYqJEYvIiIjRjAiIipGMCEiIjcjRi4=

convert(g6,parfrac,x);

KiZJInhHNiIiIiIsJiokRiMiIiNGJSIiKkYlISIi

g7 := (x^7 - 1)/(x^2 + 5)^4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnN0dGKComLCYqJEkieEdGKCIiKCIiIiEiIkYzRjMsJiokRjEiIiNGMyIiJkYzISIlNyNGLg==

convert(g7,parfrac,x);

LCoqJkkieEc2IiIiIiwmKiRGJCIiI0YmIiImRiYhIiJGJiomRiRGJkYnISIjISM6KiYsJkYkISREIkYrRiZGJkYnISIlRiYqJkYkRiZGJyEiJCIjdg==

g8 := 1/(x^10 - 3*x^8 + 8*x^7 - 18*x^6 + 24*x^5 - 26*x^4 + 24*x^3 - 15*x^2 + 8*x - 3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnOEdGKCokLDYqJEkieEdGKCIjNSIiIiokRjEiIikhIiQqJEYxIiIoRjUqJEYxIiInISM9KiRGMSIiJiIjQyokRjEiIiUhI0UqJEYxIiIkRj4qJEYxIiIjISM6RjFGNUY2RjMhIiI3I0Yu

factor(denom(g8));

KigsJkkieEc2IiIiIiIiJEYmRiYsJkYkRiYhIiJGJkYnLCYqJEYkIiIjRiZGJkYmRic=

convert(g8,parfrac,x);

LDAqJiwmSSJ4RzYiISIjIiIiRihGKCwmKiRGJSIiI0YoRihGKCEiJCNGKCIjPyokLCZGJUYoISIiRihGJyMhIzgiJEciKiYsJkYlISIpRjFGKEYoRilGJyNGKCIjXSokLCZGJUYoIiIkRihGMSNGMSImK1MnKiRGMEYxIyIjJikiJDcmKiRGMEYsI0YoIiNLKiYsJkYlISRLJCEkSCJGKEYoRilGMSNGKCIlKz8=

g9 := (x^7 + x^5 + x^3 + 1) / (x^10 - 3*x^8 + 8*x^7 - 18*x^6 + 24*x^5 - 26*x^4 + 24*x^3 - 15*x^2 + 8*x - 3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnOUdGKComLCoqJEkieEdGKCIiKCIiIiokRjEiIiZGMyokRjEiIiRGM0YzRjNGMyw2KiRGMSIjNUYzKiRGMSIiKSEiJEYwRjwqJEYxIiInISM9RjQiI0MqJEYxIiIlISNFRjZGQSokRjEiIiMhIzpGMUY8Rj1GMyEiIjcjRi4=

factor(denom(g9));

KigsJkkieEc2IiIiIiIiJEYmRiYsJkYkRiYhIiJGJkYnLCYqJEYkIiIjRiZGJkYmRic=

convert(g9,parfrac,x);

LDAqJiwmSSJ4RzYiISIkISIiIiIiRiksJiokRiUiIiNGKUYpRilGJyNGKSIjPyokLCZGJUYpRihGKSEiIyNGKSIjOyomLCZGJUYoIiIkRilGKUYqRjEjRikiI0QqJCwmRiVGKUY2RilGKCMiJDIkIiUrISkqJEYwRigjIiM4IiNrKiRGMEYnI0YpIiIpKiYsJkYlISQkWyEkLCRGKUYpRipGKCNGKSIlKz8=

Os denominadores das fra\303\247\303\265es parciais s\303\243o fatores (ou pot\303\252ncias de fatores) do denominador da fra\303\247\303\243o original. Se um fator aparece somente uma vez no denominador, ent\303\243o haver\303\241 uma \303\272nica fra\303\247\303\243o parcial com este fator no denominador; se um fator estiver repetido, ent\303\243o haver\303\241 fra\303\247\303\265es parciais com a pot\303\252ncia deste fator e com todas as pot\303\252ncias de ordem inferior em seus denominadores.

Quanto aos numeradores das fra\303\247\303\265es parciais: qualquer fra\303\247\303\243o parcial origin\303\241ria de um fator linear possui uma constante como numerador (o Maple insere esta constante defronte a fra\303\247\303\243o e considera o numerador igual a 1); qualquer fra\303\247\303\243o parcial origin\303\241ria de um fator quadr\303\241tico irredut\303\255vel possui, no m\303\241ximo, um polin\303\264mio linear no numerador.

<Text-field style="Heading 1" layout="Heading 1">Termos origin\303\241rios de Fatores Lineares</Text-field>

Resolvida a Quest\303\243o 1, na se\303\247\303\243o anterior, pode-se constatar que (al\303\251m de um poss\303\255vel polin\303\264mio) apenas dois tipos de fra\303\247\303\265es parciais aparecem na decomposi\303\247\303\243o de uma fun\303\247\303\243o racional. Se o denominador da fun\303\247\303\243o racional original contiver um fator linear 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, possivelmente repetido, haver\303\241 fra\303\247\303\265es parciais da forma 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 na decomposi\303\247\303\243o; se o denominador original contiver um fator quadr\303\241tico irredut\303\255vel 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, possivelmente repetido, haver\303\241 fra\303\247\303\265es parciais da forma 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 . Em cada caso, o maior valor de n deve ser o n\303\272mero de vezes que o fator correspondente \303\251 repetido no denominador original. Quando existem anti-derivadas para cada uma destas express\303\265es, ent\303\243o \303\251 poss\303\255vel anti-diferenciar a fun\303\247\303\243o racional.

\303\211 mais simples encontrar anti-derivadas para fra\303\247\303\265es parciais origin\303\241rias de fatores lineares: basta substituir 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 . Apresentamos a seguir dois exemplos, nos quais utilizamos o comando changevar, do pacote student.

with(student):

p1 := Int(1/(2*x - 3), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQsJkkieEdGKCIiIyEiJCIiIiEiIkYzNyNGLg==

p2 := changevar(u=2*x - 3, p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiIiMiIiIiIiNGMzcjRi4=

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJEkidUdGKCEiIkYzIyIiIiIiIzcjRi4=

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkLUkjbG5HRiU2I0kidUdGKCMiIiIiIiM3I0Yu

p5 := subs(u = 2*x - 3, p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkLUkjbG5HRiU2IywmSSJ4R0YoIiIjISIkIiIiI0Y2RjQ3I0Yu

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJCwmSSJ4R0YoIiIjISIkIiIiISIiRjIsJi1JI2xuR0YlNiNGMSNGNUYzSSJDR0YoRjVGKw==

q1 := Int(1/(4*x + 17)^6, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiQsJkkieEdGKCIiJSIjPCIiIiEiJ0YzNyNGLg==

q2 := changevar(u = 4*x + 17, q1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiJyMiIiIiIiVGMzcjRi4=

q3 := simplify(q2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwkLUkkSW50R0YnNiQqJEkidUdGKCEiJ0YzIyIiIiIiJTcjRi4=

q4 := value(q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwkKiRJInVHRighIiYjISIiIiM/NyNGLg==

q5 := subs(u = 4*x + 17, q4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKCwkKiQsJkkieEdGKCIiJSIjPCIiIiEiJiMhIiIiIz83I0Yu

q1 = q5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJCwmSSJ4R0YoIiIlIiM8IiIiISInRjIsJiokRjEhIiYjISIiIiM/SSJDR0YoRjVGKw==

Feita a substitui\303\247\303\243o indicada, o problema reduz-se a calcular 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 . Se n = 1, obt\303\251m-se um termo logar\303\255tmo; caso contr\303\241rio, a anti-derivada \303\251, novamente, uma fun\303\247\303\243o racional.

<Text-field style="Heading 2" layout="Heading 2">Quest\303\243o 2</Text-field>

Utilizar o m\303\251todo apresentado acima para encontrar as seguintes anti-derivadas:

(a) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1JJm1mcmFjR0YkNigtRiM2Iy1JI21uR0YkNiRRIjFGJ0Y4LUYjNiVGKy1GIzYjLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiZGKy1GIzYlLUZYNiRRIjRGJ0Y4LUYyNi5RMSZJbnZpc2libGVUaW1lcztGJy9GNlEoWzAsMCwwXUYnRjhGPkZBRkNGRUZHRklGS0ZNRlAtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy1GMjYtUSIrRidGOEY+RkFGQ0ZFRkdGSUZLL0ZOUSwwLjIyMjIyMjJlbUYnL0ZRRmdwLUZYNiRRIjVGJ0Y4RjgtRlg2JFEiM0YnRjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRisvJS5saW5ldGhpY2tuZXNzR0ZaLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZxLyUpYmV2ZWxsZWRHRkBGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zgci8lKmxpbmVicmVha0dRJWF1dG9GJy1GMjYvUTAmRGlmZmVyZW50aWFsRDtGJ0Y1RjhGO0Y+RkFGQ0ZFRkdGSUZLRk1GUEZbcEYr

Solu\303\247\303\243o.

p1 := Int(1/(4*x + 5)^3, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQsJkkieEdGKCIiJSIiJiIiIiEiJEYzNyNGLg==

p2 := changevar(u = 4*x + 5, p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiJCMiIiIiIiVGMzcjRi4=

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJEkidUdGKCEiJEYzIyIiIiIiJTcjRi4=

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiRJInVHRighIiMjISIiIiIpNyNGLg==

p5 := subs(u = 4*x + 5, p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkKiQsJkkieEdGKCIiJSIiJiIiIiEiIyMhIiIiIik3I0Yu

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJCwmSSJ4R0YoIiIlIiImIiIiISIkRjIsJiokRjEhIiMjISIiIiIpSSJDR0YoRjVGKw==

(b) 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

Solu\303\247\303\243o.

q1 := Int(1/(sqrt(2)*x - 1), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiQsJiomIiIjIyIiIkY0SSJ4R0YoRjZGNiEiIkY2RjhGNzcjRi4=

q2 := changevar(u = sqrt(2)*x - 1, q1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKC1JJEludEdGJzYkLCQqJkkidUdGKCEiIiIiIyMiIiJGNUY2RjM3I0Yu

q3 := simplify(q2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwkKiYiIiMjIiIiRjAtSSRJbnRHRic2JCokSSJ1R0YoISIiRjdGMkYxNyNGLg==

q4 := value(q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwkKiYiIiMjIiIiRjAtSSNsbkdGJTYjSSJ1R0YoRjJGMTcjRi4=

q5 := subs(u = sqrt(2)*x - 1, q4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKCwkKiYiIiMjIiIiRjAtSSNsbkdGJTYjLCYqJkYwRjFJInhHRihGMkYyISIiRjJGMkYxNyNGLg==

q1 = q5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJCwmKiYiIiMjIiIiRjNJInhHRihGNUY1ISIiRjVGN0Y2LCYqJkYzRjQtSSNsbkdGJTYjRjFGNUY0SSJDR0YoRjVGKw==

(c) 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

Solu\303\247\303\243o.

r1 := Int(5/(x + 2), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMUdGKC1JJEludEdGJzYkLCQqJCwmSSJ4R0YoIiIiIiIjRjUhIiIiIiZGNDcjRi4=

r2 := changevar(u = x+2, r1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMkdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiIiIiJkYzNyNGLg==

r3 := simplify(r2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyM0dGKCwkLUkkSW50R0YnNiQqJEkidUdGKCEiIkYzIiImNyNGLg==

r4 := value(r3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNEdGKCwkLUkjbG5HRiU2I0kidUdGKCIiJjcjRi4=

r5 := subs(u = x+2, r4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNUdGKCwkLUkjbG5HRiU2IywmSSJ4R0YoIiIiIiIjRjQiIiY3I0Yu

r1 = r5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQsJCokLCZJInhHRigiIiIiIiNGNCEiIiIiJkYzLCYtSSNsbkdGJTYjRjJGN0kiQ0dGKEY0Ris=

<Text-field style="Heading 1" layout="Heading 1">Termos origin\303\241rios de Fatores Quadr\303\241ticos</Text-field>

Nesta se\303\247\303\243o, estudaremos como anti-diferenciar o segundo tipo de fra\303\247\303\243o parcial, da forma 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 , originado a partir de um fator quadr\303\241tico irredut\303\255vel, 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 , no denominador original. Iniciaremos examinando dois casos especiais. O caso geral pode ser tratado a partir da combina\303\247\303\243o destes dois casos.

CASO 1: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw== . Neste caso, a fra\303\247\303\243o parcial \303\251 do tipo 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 . A constante B no numerador pode ser mantida fora do sinal de anti-derivada, de forma que pode-se assumir B = 1. Tamb\303\251m \303\251 poss\303\255vel fatorar o coeficiente a, deixando-o fora do denominador (este torna-se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJuRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== quando elevado a n-\303\251sima pot\303\252ncia). Da mesma forma, pode-se assumir a = 1. O problema reduz-se ent\303\243o a encontrar 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 , onde 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 \303\251 irredut\303\255vel. Para resolver este problema, deve-se re-escrever 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 completando o quadrado, e realizar uma substitui\303\247\303\243o adequada. \303\211 desnecess\303\241rio dizer que o Maple possui o comando completesquare. Este comando encontra-se no pacote student, que deve ser carregado caso ainda n\303\243o tenha sido feito.

Apresentamos a seguir mais dois exemplos do caso 1:

p1 := Int(1/(x^2 - 6*x + 18), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQsKCokSSJ4R0YoIiIjIiIiRjQhIiciIz1GNiEiIkY0NyNGLg==

den1 := x^2 - 6*x + 18;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkZW4xR0YoLCgqJEkieEdGKCIiIyIiIkYwISInIiM9RjI3I0Yu

den2 := completesquare(den1, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkZW4yR0YoLCYqJCwmSSJ4R0YoIiIiISIkRjIiIiNGMiIiKkYyNyNGLg==

p2 := subs(den1=den2, p1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiQsJiokLCZJInhHRigiIiIhIiRGNiIiI0Y2IiIqRjYhIiJGNTcjRi4=

p3 := changevar(x-3 = 3*tan(u), p2, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKC1JJEludEdGJzYkKiYsJiIiIkYzKiQtSSR0YW5HRiU2I0kidUdGKCIiI0YzRjMsJiIiJEYzRjRGOyEiIkY4NyNGLg==

p4 := simplify(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkLUkkSW50R0YnNiQiIiJJInVHRigjRjIiIiQ3I0Yu

p5 := value(p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkSSJ1R0YoIyIiIiIiJDcjRi4=

p6 := subs(u = arctan((x-3)/3), p5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNkdGKCwkLUknYXJjdGFuR0YlNiMsJkkieEdGKCMiIiIiIiQhIiJGNUY0NyNGLg==

p1 = p6 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJCwoKiRJInhHRigiIiMiIiJGMyEiJyIjPUY1ISIiRjMsJi1JJ2FyY3RhbkdGJTYjLCZGMyNGNSIiJEY4RjVGPkkiQ0dGKEY1Ris=

q1 := Int(1/(x^2 - 6*x + 18)^3, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiQsKCokSSJ4R0YoIiIjIiIiRjQhIiciIz1GNiEiJEY0NyNGLg==

q2 := subs(den1 = den2, q1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKC1JJEludEdGJzYkKiQsJiokLCZJInhHRigiIiIhIiRGNiIiI0Y2IiIqRjZGN0Y1NyNGLg==

q3 := changevar(x-3 = 3*tan(u), q2, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKC1JJEludEdGJzYkLCQqJCwmIiIiRjQqJC1JJHRhbkdGJTYjSSJ1R0YoIiIjRjQhIiMjRjQiJFYjRjk3I0Yu

q4 := simplify(q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSJ1R0YoIiIlRjYjIiIiIiRWIzcjRi4=

Como o Maple recusa-se a fazer a pr\303\263xima simplifica\303\247\303\243o, \303\251 preciso usar um pequeno truque.

q5 := subs(1 + tan(u)^2 = sec(u)^2, q4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSJ1R0YoIiIlRjYjIiIiIiRWIzcjRi4=

q6 := simplify(q5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNkdGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSJ1R0YoIiIlRjYjIiIiIiRWIzcjRi4=

Agora a substitui\303\247\303\243o pode ser realizada: tem-se uma express\303\243o com pot\303\252ncias de cosenos. Neste caso, deixaremos que o Maple resolva esta anti-derivada.

q7 := value(q6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxN0dGKCwmKiYsJiokLUkkY29zR0YlNiNJInVHRigiIiQiIiJGMiNGNiIiI0Y3LUkkc2luR0YlRjRGNyNGNyIkcypGNSNGNyIkWyc3I0Yu

q8 := subs(u = arctan((x-3)/3), q7);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxOEdGKCwmKiYsJiokLUkkY29zR0YlNiMtSSdhcmN0YW5HRiU2IywmSSJ4R0YoIyIiIiIiJCEiIkY7RjxGO0YyI0Y8IiIjRjstSSRzaW5HRiVGNEY7I0Y7IiRzKkY1I0Y7IiRbJzcjRi4=

q9 := simplify(q8);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxOUdGKCwkKiYsLComLUknYXJjdGFuR0YlNiMsJkkieEdGKCMiIiIiIiQhIiJGOEY4LCgqJEY2IiIjRjhGNiEiJyIjPUY4Rj1GOCokRjZGOUY5RjwhI0ZGNiIkRSIhJDsjRjhGOEY7ISIjI0Y4IiRbJzcjRi4=

<Text-field style="Heading 2" layout="Heading 2">Quest\303\243o 3</Text-field>

Utilizar o m\303\251todo acima para encontrar as anti-derivadas a seguir:

(a) 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

Solu\303\247\303\243o.

expr := 1/(x^2 + 2*x + 11);

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p1 := Int(expr, x);

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den1 := denom(expr);

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den2 := completesquare(den1);

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p2 := subs(den1 = den2, p1);

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p3 := changevar(x+1 = sqrt(10)*tan(u), p2, u);

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p4 := simplify(p3);

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p5 := value(p4);

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p6 := subs(u = arctan((x+1)/sqrt(10)), p5);

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p1 = p6 + C;

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A resposta pode ser facilmente verificada.

check := diff(p6, x);

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simplify(check);

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(b) 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

Solu\303\247\303\243o.

expr := 1/(x^2 + 6*x + 11);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoKiQsKCokSSJ4R0YoIiIjIiIiRjEiIiciIzZGMyEiIjcjRi4=

q1 := Int(expr, x);

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den1 := denom(expr);

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den2 := completesquare(den1);

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q2 := subs(den1 = den2, q1);

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q3 := changevar(x + 3 = sqrt(2)*tan(u), q2, u);

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q4 := simplify(q3);

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q5 := value(q4);

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q6 := subs(u = arctan((x+3)/sqrt(2)), q5);

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q1 = q6 + C;

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(c) 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

Solu\303\247\303\243o.

expr := 1/(2*x^2 - 4*x + 10);

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r1 := Int(expr^2, x);

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den1 := denom(expr);

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den2 := completesquare(den1);

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r2 := subs(den1 = den2, r1);

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r3 := changevar(x-1 = 2*tan(u), r2, u);

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r4 := simplify(r3);

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r5 := subs(1 + tan(u)^2 = sec(u)^2, r4);

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r6 := simplify(r5);

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r7 := value(r6);

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r8 := subs(u = arctan((x-1)/2), r7);

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r9 := simplify(r8);

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r1 = r9 + C;

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Como este \303\251 o exemplo mais complicado apresentado at\303\251 ent\303\243o, verificaremos a resposta:

check := diff(r9, x);

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simplify(check);

LCQqJCwoKiRJInhHNiIiIiMiIiJGJiEiIyIiJkYpRiojRikiIiU=

Esta \303\251 a fun\303\247\303\243o racional original, embora escrita de uma forma ligeiramente diferente.

CASO 2: 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 . Neste caso, o numerador \303\251 exatamente a derivada do polin\303\264mio irredut\303\255vel 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. Quando esta not\303\241vel coincid\303\252ncia ocorrer, deve-se efetuar a substitui\303\247\303\243o 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.

p1 := Int((2*x + 2)/(x^2 + 2*x + 11), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYsJkkieEdGKCIiI0Y0IiIiRjUsKCokRjNGNEY1RjNGNCIjNkY1ISIiRjM3I0Yu

p2 := changevar(u = x^2 + 2*x + 11, p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiRJInVHRighIiJGMjcjRi4=

p3 := value(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKC1JI2xuR0YlNiNJInVHRig3I0Yu

p4 := subs(u = x^2 + 2*x + 11, p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKC1JI2xuR0YlNiMsKCokSSJ4R0YoIiIjIiIiRjNGNCIjNkY1NyNGLg==

p1 = p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiwmSSJ4R0YoIiIjRjMiIiJGNCwoKiRGMkYzRjRGMkYzIiM2RjQhIiJGMiwmLUkjbG5HRiU2I0Y1RjRJIkNHRihGNEYr

q1 := Int((6*x + 30)/(3*x^2 + 30*x + 33)^7, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiYsJkkieEdGKCIiJyIjSSIiIkY2LCgqJEYzIiIjIiIkRjNGNSIjTEY2ISIoRjM3I0Yu

q2 := changevar(u = 3*x^2 + 30*x + 33, q1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKC1JJEludEdGJzYkKiRJInVHRighIihGMjcjRi4=

q3 := value(q2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwkKiRJInVHRighIicjISIiIiInNyNGLg==

q4 := subs(u = 3*x^2 + 30*x + 33, q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwkKiQsKCokSSJ4R0YoIiIjIiIkRjIiI0kiI0wiIiIhIicjISIiIiInNyNGLg==

q1 = q4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiwmSSJ4R0YoIiInIiNJIiIiRjUsKCokRjIiIiMiIiRGMkY0IiNMRjUhIihGMiwmKiRGNiEiJyMhIiJGM0kiQ0dGKEY1Ris=

Embora o caso 2 seja um caso muito especial, \303\251 bem simples.

O CASO GERAL.

Como mencionado anteriormente, o caso geral pode ser escrito como uma combina\303\247\303\243o dos dois casos especiais. Isto pode ser feito reescrevendo-se o numerador LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtRiw2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNTZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUTEmSW52aXNpYmxlVGltZXM7RicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZZLUYsNiZRInhGJ0Y2RjlGPC1GQDYuUSIrRidGOUZGRkhGS0ZNRk9GUUZTRlUvRlhRLDAuMjIyMjIyMmVtRicvRmVuRl1vLUYsNiZRIkJGJ0Y2RjlGPEYr como 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 (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) + (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) . Esta express\303\243o nem sempre \303\251 muito compreens\303\255vel: na verdade, a derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtRiw2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNTZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUTEmSW52aXNpYmxlVGltZXM7RicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZZLUYjNiMtSSVtc3VwR0YkNiUtRiw2JlEieEYnRjZGOUY8LUkjbW5HRiQ2JVEiMkYnRjlGRi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKy1GQDYuUSIrRidGOUZGRkhGS0ZNRk9GUUZTRlUvRlhRLDAuMjIyMjIyMmVtRicvRmVuRmlvLUYjNiUtRiw2JlEiYkYnRjZGOUY8Rj9GW29GZW8tRiw2JlEiY0YnRjZGOUY8Ris= \303\251 escrita no numerador para que se possa usar o Caso 2. Sendo assim, basta ajustar as constantes para que se tenha o numerador correto. Isto \303\251 o mesmo que dividir LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtRiw2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNTZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUTEmSW52aXNpYmxlVGltZXM7RicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZZLUYsNiZRInhGJ0Y2RjlGPC1GQDYuUSIrRidGOUZGRkhGS0ZNRk9GUUZTRlUvRlhRLDAuMjIyMjIyMmVtRicvRmVuRl1vLUYsNiZRIkJGJ0Y2RjlGPEYr por 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. Os comandos do Maple rem e quo podem ser usados para tal. Por exemplo:

expr1 := (5*x - 3)/(x^2 + 2*x + 11);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByMUdGKComLCZJInhHRigiIiYhIiQiIiJGMywoKiRGMCIiI0YzRjBGNiIjNkYzISIiNyNGLg==

Calculamos a seguir a derivada do denominador. O quociente e o resto podem ser obtidos a partir da divis\303\243o do numerador pela derivada do denominador.

d := diff(denom(expr1), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoLCZJInhHRigiIiNGMCIiIjcjRi4=

q := quo(numer(expr1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJxR0YoIyIiJiIiIzcjRi4=

r := rem(numer(expr1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoISIpNyNGLg==

expr2 := q*d/denom(expr1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByMkdGKCwkKiYsJkkieEdGKCIiI0YyIiIiRjMsKCokRjFGMkYzRjFGMiIjNkYzISIiIyIiJkYyNyNGLg==

expr3 := r/denom(expr1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByM0dGKCwkKiQsKCokSSJ4R0YoIiIjIiIiRjJGMyIjNkY0ISIiISIpNyNGLg==

Pode-se facilmente verificar que a express\303\243o original \303\251 a soma destas duas:

simplify(expr2 + expr3);

KiYsJkkieEc2IiIiJiEiJCIiIkYoLCgqJEYkIiIjRihGJEYrIiM2RighIiI=

Como expr2 \303\251 um exemplo do Caso 2 (exceto pelo fator 5/2, que n\303\243o agrega nenhuma dificuldade adicional) e expr3 \303\251 um exemplo do Caso 1, pode-se encontrar as anti-derivadas para cada express\303\243o e, portanto, para a express\303\243o original. Da mesma forma, se tiv\303\251ssemos iniciado com uma pot\303\252ncia de 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 no denominador, por exemplo,

Since expr2 is an example of Case 2 (apart from the factor of 5/2, which causes no difficulty), and expr3 is an example of Case 1, we can find anti-derivatives for each of them, and hence for the original expression. In exactly the same way, if we had started with a power of 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 in the denominator, for example

expr4 := (5*x - 3)/(x^2 + 2*x + 11)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByNEdGKComLCZJInhHRigiIiYhIiQiIiJGMywoKiRGMCIiI0YzRjBGNiIjNkYzRjI3I0Yu

tal express\303\243o poderia ser re-escrita como a soma de duas express\303\265es:

expr5 := q*d/denom(expr1)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByNUdGKCwkKiYsJkkieEdGKCIiI0YyIiIiRjMsKCokRjFGMkYzRjFGMiIjNkYzISIkIyIiJkYyNyNGLg==

expr6 := r/denom(expr1)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByNkdGKCwkKiQsKCokSSJ4R0YoIiIjIiIiRjJGMyIjNkY0ISIkISIpNyNGLg==

Verifica\303\247\303\243o:

simplify(expr5 + expr6);

KiYsJkkieEc2IiIiJiEiJCIiIkYoLCgqJEYkIiIjRihGJEYrIiM2RihGJw==

Finalmente, pode-se anti-diferenciar qualquer fra\303\247\303\243o parcial origin\303\241ria de um fator quadr\303\241tico irredut\303\255vel e, portanto, qualquer fun\303\247\303\243o racional.

<Text-field style="Heading 2" layout="Heading 2">Quest\303\243o 4</Text-field>

Re-escrever cada uma das express\303\265es a seguir como a soma de uma express\303\243o do Caso 1 e outra do Caso 2.

p1 := (3*x + 4)/(x^2 + x + 1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKComLCZJInhHRigiIiQiIiUiIiJGMywoKiRGMCIiI0YzRjBGM0YzRjMhIiI3I0Yu

Solu\303\247\303\243o. (Manualmente, este procedimento pode ser ainda mais simples. Contudo, apresentamos a seguir solu\303\247\303\265es do Maple, segundo o m\303\251todo indicado acima.)

den := denom(p1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkZW5HRigsKCokSSJ4R0YoIiIjIiIiRjBGMkYyRjI3I0Yu

d := diff(den, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoLCZJInhHRigiIiMiIiJGMTcjRi4=

q := quo(numer(p1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJxR0YoIyIiJCIiIzcjRi4=

r := rem(numer(p1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoIyIiJiIiIzcjRi4=

p2 := q*d/den + r/den;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKCwmKiYsJkkieEdGKCIiIyIiIkYzRjMsKCokRjFGMkYzRjFGM0YzRjMhIiIjIiIkRjIqJEY0RjYjIiImRjI3I0Yu

q1 := (6*x + 5)/(x^2 + 4*x + 13)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKComLCZJInhHRigiIiciIiYiIiJGMywoKiRGMCIiI0YzRjAiIiUiIzhGMyEiIzcjRi4=

Solu\303\247\303\243o.

quad := x^2 + 4*x + 13;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVxdWFkR0YoLCgqJEkieEdGKCIiIyIiIkYwIiIlIiM4RjI3I0Yu

d := diff(quad, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoLCZJInhHRigiIiMiIiUiIiI3I0Yu

q := quo(numer(q1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJxR0YoIiIkNyNGLg==

r := rem(numer(q1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoISIoNyNGLg==

q2 := (q*d)/quad^2 + r/quad^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKCwmKiYsJkkieEdGKCIiIyIiJSIiIkY0LCgqJEYxRjJGNEYxRjMiIzhGNCEiIyIiJCokRjVGOCEiKDcjRi4=

<Text-field style="Heading 2" layout="Heading 2">Quest\303\243o 5</Text-field>

Utilizar o procedimento descrito acima para encontrar as anti-derivadas das express\303\265es a seguir: utilizar inicialmente o comando convert para encontrar a decomposi\303\247\303\243o por fra\303\247\303\265es parciais e, em seguida, calcular as anti-derivadas para cada uma das fra\303\247\303\265es parciais.

p1 := 1/((x-1)*(x+1)^2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKComLCZJInhHRigiIiIhIiJGMUYyLCZGMEYxRjFGMSEiIzcjRi4=

Solu\303\247\303\243o.

p2 := convert(p1, parfrac, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKCwoKiQsJkkieEdGKCIiIkYyRjIhIiMjISIiIiIjKiQsJkYxRjJGNUYyRjUjRjIiIiUqJEYwRjUjRjVGOjcjRi4=

p3 := Int(1/(4*(x-1)), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKC1JJEludEdGJzYkKiQsJkkieEdGKCIiJSEiJSIiIiEiIkYzNyNGLg==

p4 := changevar(u=x-1, p3, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiIiMiIiIiIiVGMzcjRi4=

p5 := simplify(p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkLUkkSW50R0YnNiQqJEkidUdGKCEiIkYzIyIiIiIiJTcjRi4=

p6 := value(p5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNkdGKCwkLUkjbG5HRiU2I0kidUdGKCMiIiIiIiU3I0Yu

p7 := subs(u=x-1, p6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwN0dGKCwkLUkjbG5HRiU2IywmSSJ4R0YoIiIiISIiRjQjRjQiIiU3I0Yu

Como as duas fra\303\247\303\265es parciais a seguir envolvem a mesma substitui\303\247\303\243o, resolverem\303\264-las ao mesmo tempo.

p8 := op(2, p2) + op(3, p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwOEdGKCwmKiQsJkkieEdGKCIiIiEiIkYyRjMjRjIiIiUqJCwmRjFGMkYyRjJGMyNGM0Y1NyNGLg==

p9 := Int(p8, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwOUdGKC1JJEludEdGJzYkLCYqJCwmSSJ4R0YoIiIiISIiRjVGNiNGNSIiJSokLCZGNEY1RjVGNUY2I0Y2RjhGNDcjRi4=

p10 := changevar(u=x+1, p9, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRwMTBHRigtSSRJbnRHRic2JCwmKiQsJkkidUdGKCIiJSEiKSIiIiEiIkY3KiRGNEY4I0Y4RjVGNDcjRi4=

p11 := value(p10);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRwMTFHRigsJi1JI2xuR0YlNiMsJkkidUdGKCIiJSEiKSIiIiNGNkY0LUYwNiNGMyMhIiJGNDcjRi4=

p12 := subs(u=x+1, p11);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRwMTJHRigsJi1JI2xuR0YlNiMsJkkieEdGKCIiJSEiJSIiIiNGNkY0LUYwNiMsJkYzRjZGNkY2IyEiIkY0NyNGLg==

p13 := p7 + p12;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRwMTNHRigsKC1JI2xuR0YlNiMsJkkieEdGKCIiIiEiIkY0I0Y0IiIlLUYwNiMsJkYzRjchIiVGNEY2LUYwNiMsJkYzRjRGNEY0I0Y1Rjc3I0Yu

Int(p1, x) = p13 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiwmSSJ4R0YoIiIiISIiRjNGNCwmRjJGM0YzRjMhIiNGMiwqLUkjbG5HRiU2I0YxI0YzIiIlLUY5NiMsJkYyRjwhIiVGM0Y7LUY5NiNGNSNGNEY8SSJDR0YoRjNGKw==

q1 := (3*x + 4)/(x^2 + x + 1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKComLCZJInhHRigiIiQiIiUiIiJGMywoKiRGMCIiI0YzRjBGM0YzRjMhIiI3I0Yu

Solu\303\247\303\243o. Como o denominador \303\251 irredut\303\255vel (o que se torna evidente ap\303\263s completar-se o quadrado), esta express\303\243o j\303\241 encontra-se na forma de fra\303\247\303\243o parcial. Sendo assim, basta procecer de maneira an\303\241loga \303\240 Quest\303\243o 4.

den := denom(q1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkZW5HRigsKCokSSJ4R0YoIiIjIiIiRjBGMkYyRjI3I0Yu

d := diff(den, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoLCZJInhHRigiIiMiIiJGMTcjRi4=

q := quo(numer(q1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJxR0YoIyIiJCIiIzcjRi4=

r := rem(numer(q1), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoIyIiJiIiIzcjRi4=

q2 := q*d/den;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKCwkKiYsJkkieEdGKCIiIyIiIkYzRjMsKCokRjFGMkYzRjFGM0YzRjMhIiIjIiIkRjI3I0Yu

q3 := r/den;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwkKiQsKCokSSJ4R0YoIiIjIiIiRjJGNEY0RjQhIiIjIiImRjM3I0Yu

A express\303\243o q1 foi dividida em empress\303\265es que enquadram-se tanto no caso 1 como no caso 2. Portanto, \303\251 poss\303\255vel encontrar anti-derivadas para cada uma destas express\303\265es. Iniciaremos com q2.

q4 := Int(q2, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKC1JJEludEdGJzYkLCQqJiwmSSJ4R0YoIiIjIiIiRjZGNiwoKiRGNEY1RjZGNEY2RjZGNiEiIiMiIiRGNUY0NyNGLg==

q5 := changevar(u = x^2 + x + 1, q4, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiIiMiIiQiIiNGMzcjRi4=

q6 := value(q5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNkdGKCwkLUkjbG5HRiU2I0kidUdGKCMiIiQiIiM3I0Yu

q7 := subs(u = x^2 + x + 1, q6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxN0dGKCwkLUkjbG5HRiU2IywoKiRJInhHRigiIiMiIiJGNEY2RjZGNiMiIiRGNTcjRi4=

Para q3, tem-se:

q8 := Int(q3, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxOEdGKC1JJEludEdGJzYkLCQqJCwoKiRJInhHRigiIiMiIiJGNUY3RjdGNyEiIiMiIiZGNkY1NyNGLg==

den1 := completesquare(den);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkZW4xR0YoLCYqJCwmSSJ4R0YoIiIiI0YyIiIjRjJGNEYyIyIiJCIiJUYyNyNGLg==

q9 := subs(den=den1, q8);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxOUdGKC1JJEludEdGJzYkLCQqJCwmKiQsJkkieEdGKCIiIiNGNyIiI0Y3RjlGNyMiIiQiIiVGNyEiIiMiIiZGOUY2NyNGLg==

q10 := changevar(x + 1/2 = sqrt(3/4)*tan(u), q9, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxMTBHRigtSSRJbnRHRic2JCwkKigsJiokLUkkdGFuR0YlNiNJInVHRigiIiMjIiIkIiIlRjoiIiIhIiJGOyNGPUY5LCZGPUY9RjRGPUY9IyIiJkY8Rjg3I0Yu

q11 := simplify(q10);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxMTFHRigsJComIiIkIyIiIiIiIy1JJEludEdGJzYkRjJJInVHRihGMiMiIiZGMDcjRi4=

q12 := value(q11);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxMTJHRigsJComIiIkIyIiIiIiI0kidUdGKEYyIyIiJkYwNyNGLg==

q13 := subs(u = arctan((x + 1/2)/sqrt(3/4)), q12);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxMTNHRigsJComIiIkIyIiIiIiIy1JJ2FyY3RhbkdGJTYjLCQqJiwmSSJ4R0YoRjJGMUYyRjJGMEYxI0YzRjBGMiMiIiZGMDcjRi4=

Juntando as duas partes, obt\303\251m-se

q14 := q7 + q13;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRxMTRHRigsJi1JI2xuR0YlNiMsKCokSSJ4R0YoIiIjIiIiRjRGNkY2RjYjIiIkRjUqJkY4I0Y2RjUtSSdhcmN0YW5HRiU2IywkKiYsJkY0RjZGOkY2RjZGOEY6I0Y1RjhGNiMiIiZGODcjRi4=

Int(q1, x) = q14 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiwmSSJ4R0YoIiIkIiIlIiIiRjUsKCokRjIiIiNGNUYyRjVGNUY1ISIiRjIsKC1JI2xuR0YlNiNGNiNGM0Y4KiZGMyNGNUY4LUknYXJjdGFuR0YlNiMsJComLCZGMkY1RkBGNUY1RjNGQCNGOEYzRjUjIiImRjNJIkNHRihGNUYr

Como este \303\251 um c\303\241lculo bastante longo, tomaremos o cuidado de verificar a resposta.

check := diff(q14, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjaGVja0dGJywmKiYsJkkieEdGKCIiIyIiIkYzRjMsKCokRjFGMkYzRjFGM0YzRjMhIiIjIiIkRjIqJCwmKiQsJkYxRjMjRjNGMkYzRjIjIiIlRjhGM0YzRjYjIiM1Rjg3I0Yu

simplify(check);

KiYsJkkieEc2IiIiJCIiJSIiIkYoLCgqJEYkIiIjRihGJEYoRihGKCEiIg==

r1 := 1/(x^3 - 1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMUdGKCokLCYqJEkieEdGKCIiJCIiIiEiIkYzRjQ3I0Yu

Solu\303\247\303\243o.

r2 := convert(r1, parfrac, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMkdGKCwmKiQsJkkieEdGKCIiIiEiIkYyRjMjRjIiIiQqJiwmRjFGMyEiI0YyRjIsKCokRjEiIiNGMkYxRjJGMkYyRjNGNDcjRi4=

Pode-se agora anti-diferenciar as fra\303\247\303\265es parciais, uma a uma.

r3 := op(1, r2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyM0dGKCwkKiQsJkkieEdGKCIiIiEiIkYyRjMjRjIiIiQ3I0Yu

r4 := Int(r3, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNEdGKC1JJEludEdGJzYkLCQqJCwmSSJ4R0YoIiIiISIiRjVGNiNGNSIiJEY0NyNGLg==

r5 := changevar(u = x-1, r4, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNUdGKC1JJEludEdGJzYkLCQqJEkidUdGKCEiIiMiIiIiIiRGMzcjRi4=

r6 := value(r5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNkdGKCwkLUkjbG5HRiU2I0kidUdGKCMiIiIiIiQ3I0Yu

r7 := subs(u = x-1, r6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyN0dGKCwkLUkjbG5HRiU2IywmSSJ4R0YoIiIiISIiRjQjRjQiIiQ3I0Yu

Uma vez resolvida a primeira fra\303\247\303\243o parcial, pode-se continuar com a segunda. Neste caso, as fra\303\247\303\265es devem ser divididas de acordo com os casos 1 e 2. (r9 e r10 , abaixo).

r8 := op(2, r2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyOEdGKCwkKiYsJkkieEdGKCEiIiEiIyIiIkY0LCgqJEYxIiIjRjRGMUY0RjRGNEYyI0Y0IiIkNyNGLg==

den := denom(r8);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkZW5HRigsKCokSSJ4R0YoIiIjIiIkRjBGMkYyIiIiNyNGLg==

d := diff(den, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoLCZJInhHRigiIiciIiQiIiI3I0Yu

q := quo(numer(r8), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJxR0YoIyEiIiIiJzcjRi4=

r := rem(numer(r8), d, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoIyEiJCIiIzcjRi4=

r9 := q*d/den;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyOUdGKCwkKiYsJkkieEdGKCIiJyIiJCIiIkY0LCgqJEYxIiIjRjNGMUYzRjNGNCEiIiNGOEYyNyNGLg==

r10 := r/den;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTBHRigsJCokLCgqJEkieEdGKCIiIyIiJEYyRjRGNCIiIiEiIiMhIiRGMzcjRi4=

Anti-diferenciando r9:

r11 := Int(r9, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTFHRigtSSRJbnRHRic2JCwkKiYsJkkieEdGKCIiJyIiJCIiIkY3LCgqJEY0IiIjRjZGNEY2RjZGNyEiIiNGO0Y1RjQ3I0Yu

r12 := changevar(u = 3*x^2 + 3*x + 3, r11, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTJHRigtSSRJbnRHRic2JCwkKiRJInVHRighIiIjRjQiIidGMzcjRi4=

r13 := value(r12);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTNHRigsJC1JI2xuR0YlNiNJInVHRigjISIiIiInNyNGLg==

r14 := subs(u = 3*x^2 + 3*x + 3, r13);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTRHRigsJC1JI2xuR0YlNiMsKCokSSJ4R0YoIiIjIiIkRjRGNkY2IiIiIyEiIiIiJzcjRi4=

Anti-diferenciando r10:

r15 := Int(r10, x);

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den1 := completesquare(den);

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r16 := subs(den = den1, r15);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTZHRigtSSRJbnRHRic2JCwkKiQsJiokLCZJInhHRigiIiIjRjciIiNGN0Y5IiIkIyIiKiIiJUY3ISIiIyEiJEY5RjY3I0Yu

r17 := changevar(x + 1/2 = sqrt(3/4)*tan(u), r16, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTdHRigtSSRJbnRHRic2JCwkKigsJiokLUkkdGFuR0YlNiNJInVHRigiIiMjIiIqIiIlRjoiIiIhIiIiIiQjRj1GOSwmRj1GPUY0Rj1GPSMhIiRGPEY4NyNGLg==

r18 := simplify(r17);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMThHRigsJComIiIkIyIiIiIiIy1JJEludEdGJzYkRjJJInVHRihGMiMhIiJGMDcjRi4=

r19 := value(r18);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTlHRigsJComIiIkIyIiIiIiI0kidUdGKEYyIyEiIkYwNyNGLg==

r20 := subs(u = arctan((x + 1/2)/sqrt(3/4)), r19);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMjBHRigsJComIiIkIyIiIiIiIy1JJ2FyY3RhbkdGJTYjLCQqJiwmSSJ4R0YoRjJGMUYyRjJGMEYxI0YzRjBGMiMhIiJGMDcjRi4=

Agrupando as respostas:

r21 := r7 + r14 + r20;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMjFHRigsKC1JI2xuR0YlNiMsJkkieEdGKCIiIiEiIkY0I0Y0IiIkLUYwNiMsKCokRjMiIiNGN0YzRjdGN0Y0I0Y1IiInKiZGNyNGNEY8LUknYXJjdGFuR0YlNiMsJComLCZGM0Y0RkBGNEY0RjdGQCNGPEY3RjQjRjVGNzcjRi4=

Int(r1, x) := r21 + C;

Error, invalid left hand side of assignment

Verifica\303\247\303\243o:

check := diff(r21, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjaGVja0dGJywoKiQsJkkieEdGKCIiIiEiIkYyRjMjRjIiIiQqJiwmRjEiIidGNUYyRjIsKCokRjEiIiNGNUYxRjVGNUYyRjMjRjNGOCokLCYqJCwmRjFGMiNGMkY7RjJGOyMiIiVGNUYyRjJGMyMhIiNGNTcjRi4=

simplify(check);

KiYsJkkieEc2IiIiIiEiIkYmRicsKCokRiQiIiNGJkYkRiZGJkYmRic=

Esta \303\251 a fun\303\247\303\243o racional original, como se pode observar pela multiplica\303\247\303\243o do denominador.