C\303\241lculo IILi\303\247\303\243o 10: Integra\303\247\303\243o por Partes<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Anti-diferencia\303\247\303\243o por Partes</Font></Text-field>restart;O segundo m\303\251todo mais importante da antidiferencia\303\247\303\243o a ser estudado \303\251 a anti-diferencia\303\247\303\243o por partes. Este este m\303\251todo pode ser resumido na f\303\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 .O Maple possui um comando que faz a integra\303\247\303\243o por partes, chamado intparts. Este comando encontra-se no pacote student, que deve ser recarregado, uma vez que o comando restart removeu-o da mem\303\263ria.with(student);(Note que desta vez n\303\243o suprimimos a lista de comandos.) O comando intparts possui dois argumentos: o primeiro \303\251 a express\303\243o a ser anti-diferenciada, e o segundo \303\251 a escolha para u, a parte a ser diferenciada. N\303\243o \303\251 preciso especificar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJERicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidkYnL0Y1USV0cnVlRicvRjhRJ2l0YWxpY0YnRjdGN0Y3RitGNw==, pois este ser\303\241 simplesmente todo o restante do integrando.Como exemplo, calcularemos a integral 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. Basta olhar para o exemplo para perceber que a escolha deve ser u = x.p1 := Int(x*exp(-2*x),x);p2 := intparts(p1,x);p3 := simplify(p2);Ainda resta uma integral a ser resolvida. Por\303\251m, esta \303\251 uma integral suficientemente simples de ser resolvida. Desta forma, basta utilizar o Maple:p4 := value(p3);p1 = p4 + C;Finalmente, pode-se verificar a resposta por diferencia\303\247\303\243o:diff(p4,x);A diferencia\303\247\303\243o de p4 resulta na fun\303\247\303\243o inicial. Portanto, p4 \303\251 uma anti-derivada para esta fun\303\247\303\243o.Existem alguns truques interessantes que podem ser feitos com o comando intparts. Por exemplo, uma forma usual de se utilizar o m\303\251todo das parte para o c\303\241lculo da 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 consiste em reescrev\303\252-la como 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 , escolhendo 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.p1 := Int(ln(x),x);p2 := intparts(p1,ln(x));p3 := value(p2);p1 = p3 + C;Obviamente, a verifica\303\247\303\243o da resposta pode ser feita por diferencia\303\247\303\243o:diff(p3,x);<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Quest\303\243o 2</Font></Text-field>Utilizar o m\303\251todo descrito acima para encontrar as anti-derivadas a seguir. (Todas s\303\243o tiradas do Exerc\303\255cio 7.1 de Stewart. Aconselha-se resolv\303\252-las todas --- al\303\251m de outras --- como exerc\303\255cio.)(a) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KS1JI21vR0YkNi9RKCYjODc0NztGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZPLUYjNictRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy1GMjYtUTEmSW52aXNpYmxlVGltZXM7RidGOEY+RkFGQ0ZFRkdGSUZLRk1GUC1GIzYmLUYsNiVRJGNvc0YnL0ZYRkBGOC1GMjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y4Rj5GQUZDRkVGR0ZJRktGTUZQLUkobWZlbmNlZEdGJDYkLUYjNiRGVEY4RjhGOEYrRjhGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0ZccC8lKmxpbmVicmVha0dRJWF1dG9GJy1GMjYvUTAmRGlmZmVyZW50aWFsRDtGJ0Y1RjhGO0Y+RkFGQ0ZFRkdGSUZLRk1GUEZURjhGK0Y4
Solu\303\247\303\243o.p1 := Int(x*cos(x),x);p2 := intparts(p1,x);p3 := value(p2);p1 = p3 + C;Verifica\303\247\303\243o: (Cada uma das respostas deve ser verificada!)diff(p3,x);(b) 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
Solu\303\247\303\243o.p1 := Int(x*ln(x), x);p2 := intparts(p1, ln(x));p3 := simplify(p2);p4 := value(p3);p1 = p4 + C;Verifica\303\247\303\243o:diff(p4,x);(c) 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
Solution. Est\303\241 claro que a pot\303\252ncia de x deve ser diferenciada. Isto implicar\303\241 na diminui\303\247\303\243o da pot\303\252ncia em 1 grau, resultando assim numa integral mais simples. No entanto a integral resultante ainda n\303\243o pode ser resolvida. Para isto, \303\251 necess\303\241rio fazer a integra\303\247\303\243o por partes ainda mais uma vez.p1 := Int(x^2 * sin(2*x), x);p2 := intparts(p1, x^2);p3 := simplify(p2);p4 := intparts(p3, x);p5 := simplify(p4);p6 := value(p5);p1 = p6 + C;Verifica\303\247\303\243o:diff(p6,x);(d) 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
Solu\303\247\303\243o. Tem-se aqui o mesmo caso do problema (c): \303\251 preciso integrar por partes duas vezes.p1 := Int(x^2 * exp(-x), x);p2 := intparts(p1, x^2);p3 := simplify(p2);p4 := intparts(p3, x);p5 := simplify(p4);p6 := value(p5);p1 = p6 + C;Verifica\303\247\303\243o:diff(p6, x);<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Quest\303\243o 3</Font></Text-field>Encontrar as anti-derivadas a seguir. Ser\303\241 necess\303\241rio utilizar uma combina\303\247\303\243o de ambos os m\303\251todos: substitu\303\247\303\243o e integra\303\247\303\243o por partes.(a) 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
Solu\303\247\303\243o. Quando se tem uma fun\303\247\303\243o exponencial na integral, \303\251 aconselh\303\241vel considerar u como sendo a fun\303\247\303\243o contida no expontente, qualquer que seja esta.p1 := Int(x^3 * exp(x^2), x);p2 := changevar(u=x^2, p1, u);p3 := simplify(p2);p4 := intparts(p3, u);p5 := value(p4);p6 := subs(u=x^2, p5);p1 = p6 + C;Verifica\303\247\303\243o:diff(p6, x);(b) 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Solu\303\247\303\243o.p1 := Int(cos(x)*ln(sin(x)), x);p2 := changevar(u=sin(x), p1, u);Esta anti-derivada j\303\241 foi calculada anteriormente nesta folha de trabalho. Por\303\251m, resolveremos novamente o problema para praticar (e porque este n\303\243o se encontra na lista da p. 416).p3 := intparts(p2, ln(u));p4 := value(p3);p5 := subs(u=sin(x), p4);p1 = p5 + C;Verifica\303\247\303\243o:diff(p5, x);