C\303\241lculo IILi\303\247\303\243o 13: Integrais Impr\303\263prias-Parte 2 At\303\251 ent\303\243o, em nosso estudo de integra\303\247\303\243o, temos considerado 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 , sendo f \303\251 uma fun\303\247\303\243o no intervalo limitado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=. Deseja-se agora analisar o que acontece quando ambos, f ou o intervaloLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic= , tornam-se ilimitados. Em ambos os casos, tem-se uma integral impr\303\263pria (as integrais estudadas at\303\251 agora s\303\243o integrais pr\303\263prias). Como veremos mais adiante, uma integral impr\303\263pria n\303\243o \303\251 definida diretamente em termos de parti\303\247\303\265es e somas, mas sim como o limite de integrais pr\303\263prias.
<Text-field style="Heading 1" layout="Heading 1">Tipo 1: Intervalos Infinitos</Text-field> Iniciaremos com o caso onde o intervalo de integra\303\247\303\243o, [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEoJmluZmluO0YnL0YwRj1GOQ==), \303\251 ilimitado \303\240 esquerda. Neste caso, definimos 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 , contanto que o limite exista. Se f \303\251 uma fun\303\247\303\243o positiva, este limite pode ser interpretado como sendo a \303\241rea total sob o gr\303\241fico de f, \303\240 direita de a; em geral, tem-se as mesmas interpreta\303\247\303\265es que para a integral pr\303\263pria.O Maple \303\251 capaz de resolver muitas integrais impr\303\263prias como, por exemplo, Int(1/(2*x - 5)^3, x=4..infinity) = int(1/(2*x - 5)^3, x=4..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJCwmSSJ4R0YoIiIjISImIiIiISIkL0YyOyIiJUkpaW5maW5pdHlHRiYjRjUiI09GKw== f := x->1/(2*x - 5)^3; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQsJjkkIiIjISImIiIiISIkRihGKEYoNyNGLg== p1 := plot(f(x), x=3.5..8,thickness=2): p2 := plottools[line]([4,0],[4,f(4)],color=blue,thickness=2): plots[display](p1,p2); 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Contudo, o uso destes comando nos oferecem pouca clareza. Desta forma, resolveremos a mesma integral passo a passo. Inicialmente, \303\251 preciso resolver a integral de 4 a um limite superior vari\303\241vel b. Manualmente, \303\251 preciso encontrar uma anti-derivada para a fun\303\247\303\243o e aplicar ent\303\243o o Teorema Fundamental do C\303\241lculo: ad := int(f(x), x); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhZEdGKCwkKiQsJkkieEdGKCIiIyEiJiIiIiEiIyMhIiIiIiU3I0Yu inttob := subs(x=b, ad) - subs(x=4, ad); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2JHRigsJiokLCZJImJHRigiIiMhIiYiIiIhIiMjISIiIiIlI0Y0IiNPRjQ3I0Yu Pode-se obter o mesmo resuldado diretamente com o Maple, no c\303\241lculo da integral de 4 at\303\251 b. (A resposta aparece de forma diferente. Por\303\251m, verifica-se que \303\251 a mesma.) int(f(x), x=4..b); Warning, unable to determine if 5/2 is between 4 and b; try to use assumptions or use the AllSolutions option LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRpbnRHRiU2JCokLCZJInhHRigiIiMhIiYiIiIhIiQvRjE7IiIlSSJiR0YoRis= simplify(inttob - int(f(x), x=4..b)); Warning, unable to determine if 5/2 is between 4 and b; try to use assumptions or use the AllSolutions option LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKCokLCZJImJHRigiIiMhIiYiIiIhIiMjISIiIiIlI0YyIiNPRjItSSRpbnRHRiU2JCokLCZJInhHRihGMEYxRjIhIiQvRj47RjZGL0Y1Ris= Tendo encontrado a integral at\303\251 b, de uma forma ou de outra, pode-se ent\303\243o tormar o limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnL0Y1RkJGPkYr : Int(1/(2*x - 5)^3, x=4..infinity) = limit(inttob, b=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJCwmSSJ4R0YoIiIjISImIiIiISIkL0YyOyIiJUkpaW5maW5pdHlHRiYjRjUiI09GKw== Esta resposta encontra-se em acordo com a resposta original do Maple.Se o intervalo for ilimitado \303\240 esquerda, procedemos de forma similar, definindo 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 . Int(exp(3*x), x=-infinity..2) = int(exp(3*x), x=-infinity..2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQtSSRleHBHRiU2IywkSSJ4R0YoIiIkL0Y0OywkSSlpbmZpbml0eUdGJiEiIiIiIywkLUYxNiMiIicjIiIiRjVGKw== inttoa := int(exp(3*x), x=a..2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2FHRigsJi1JJGV4cEdGJTYjLCRJImFHRigiIiQjISIiRjQtRjA2IyIiJyMiIiJGNDcjRi4= limit(inttoa, a=-infinity); LCQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiJyMiIiIiIiQ=
Finalmente, se a integral \303\251 ilimitada de ambos os lados, escolhe-se um ponto conveniente, c (em geral, c = 0) e define-se
<Text-field style="Heading 1" layout="Heading 1"></Text-field> 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 , contanto que ambas as integrais do lado direito existam. Esta condi\303\247\303\243o \303\251 crucial. Consideremos, por exemplo, a integral f := x->(1 + (1+x^2)*sin(x))/(1 + x^2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiYsJiIiIkY2KiYsJkY2RjYqJDkkIiIjRjZGNi1JJHNpbkdGJTYjRjpGNkY2RjZGOCEiIkYoRihGKDcjRi4= Int(f(x), x=-infinity..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JComLCYiIiJGMSomLCYqJEkieEdGKCIiI0YxRjFGMUYxLUkkc2luR0YlNiNGNUYxRjFGMUYzISIiL0Y1OywkSSlpbmZpbml0eUdGJkY6Rj5GKw== plot(f(x), x=-50..50, thickness=2); 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O gr\303\241fico indica que a integral sobre todo o eixo real pode n\303\243o existir. Deve-se ter um m\303\255nimo de cuidado, para que n\303\243o se calcule esta integral como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JJ211bmRlckdGJDYlLUkjbW9HRiQ2L1EkbGltRicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2JS1GLDYlUSJhRicvJSdpdGFsaWNHRk4vRjxRJ2l0YWxpY0YnLUY1Ni1RJyZyYXJyO0YnRjtGQUZERkZGSEZKL0ZNRkNGT0ZRL0ZVRlMtRiw2JVEoJmluZmluO0YnL0ZnbkZDRjsvJSxhY2NlbnR1bmRlckdGQ0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmpvLyUqbGluZWJyZWFrR1ElYXV0b0YnLUkobWZlbmNlZEdGJDYkLUYjNigtSShtc3Vic3VwR0YkNictRjU2L1EmJmludDtGJ0Y4RjtGPkZBRkRGRkZIRkpGXW9GT0ZRRl5vLUYjNiQtRjU2LVEqJnVtaW51czA7RidGO0ZBRkRGRkZIRkpGXW9GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZkcUZZRlkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSZtZnJhY0dGJDYoLUYjNiVGKy1GIzYmLUkjbW5HRiQ2JFEiMUYnRjstRjU2LVEiK0YnRjtGQUZERkZGSEZKRl1vRk9GY3FGZXEtRiM2Ji1GZHA2JC1GIzYmRmNyRmdyLUYjNiMtSSVtc3VwR0YkNiUtRiw2JVEieEYnRmZuRmhuLUZkcjYkRmhxRjsvRmdxRltyRitGOy1GNTYuUTEmSW52aXNpYmxlVGltZXM7RicvRjlRKFswLDAsMF1GJ0Y7RkFGREZGRkhGSkZdb0ZPRlFGXm8tRiM2JS1GLDYlUSRzaW5GJ0Zib0Y7LUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjtGQUZERkZGSEZKRl1vRk9GUUZeby1GZHA2JC1GIzYjRmVzRjtGK0YrRistRiM2JUYrRl5zRisvJS5saW5ldGhpY2tuZXNzR0Zmci8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZidS8lKWJldmVsbGVkR0ZDRistRmZvNiZGaG8vRlxwUSYwLjNlbUYnRl5wRmBwLUY1Ni9RMCZEaWZmZXJlbnRpYWxEO0YnRjhGO0Y+RkFGREZGRkhGSkZdb0ZPRlFGXm9GZXNGO0Yr finiteint := int((1 + (1+x^2)*sin(x))/(1+x^2),x = -a .. a); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpmaW5pdGVpbnRHRigsJC1JJ2FyY3RhbkdGJTYjSSJhR0YoIiIjNyNGLg== Int(f(x) ,x = -infinity..infinity) = limit(finiteint, a=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJiwmIiIiRjIqJiwmKiRJInhHRigiIiNGMkYyRjJGMi1JJHNpbkdGJTYjRjZGMkYyRjJGNCEiIi9GNjssJEkpaW5maW5pdHlHRiZGO0Y/SSNQaUdGJkYr ESTA RESPOSTA EST\303\201 ERRADA! Para calcular corretamente a integral, deve-se fazer com que as duas extremidades tendam a +/-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= separadamente. Ao escolhermos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw==, a integral impr\303\263pria at\303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= \303\251 calculada por inttob := int(f(x), x=0..b); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2JHRigsKCIiIkYvLUkkY29zR0YlNiNJImJHRighIiItSSdhcmN0YW5HRiVGMkYvNyNGLg== Int(f(x), x=0..infinity) = limit(inttob, b=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJiwmIiIiRjIqJiwmKiRJInhHRigiIiNGMkYyRjJGMi1JJHNpbkdGJTYjRjZGMkYyRjJGNCEiIi9GNjsiIiFJKWluZmluaXR5R0YmOywkSSNQaUdGJiNGMkY3LCZGN0YyRkJGQ0Yr A apar\303\252ncia um tanto estranha da resposta mostra que algo de errado aconteceu. De fato, tem-se que a express\303\243o inttob n\303\243o possui limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnL0Y1RkJGPkYr . N\303\243o h\303\241 necessidade de continuar: quando uma das integrais impr\303\263pria semi-infinitas n\303\243o existe, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GIzYkLUY1Ni1RKiZ1bWludXMwO0YnRjtGQUZERkZGSEZKRkxGTi9GUVEsMC4yMjIyMjIyZW1GJy9GVEZlbi1GLDYlUSgmaW5maW47RicvJSdpdGFsaWNHRkNGO0Znbi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JJm1mcmFjR0YkNigtRiM2JUYrLUYjNiYtSSNtbkdGJDYkUSIxRidGOy1GNTYtUSIrRidGO0ZBRkRGRkZIRkpGTEZORlpGZm4tRiM2Ji1JKG1mZW5jZWRHRiQ2JC1GIzYmRmlvRl1wLUYjNiMtSSVtc3VwR0YkNiUtRiw2JVEieEYnL0Zbb1EldHJ1ZUYnL0Y8USdpdGFsaWNGJy1Gam82JEZeb0Y7L0Zdb0Zhb0YrRjstRjU2LlExJkludmlzaWJsZVRpbWVzO0YnL0Y5UShbMCwwLDBdRidGO0ZBRkRGRkZIRkpGTEZORlBGUy1GIzYlLUYsNiVRJHNpbkYnRmpuRjstRjU2LVEwJkFwcGx5RnVuY3Rpb247RidGO0ZBRkRGRkZIRkpGTEZORlBGUy1GY3A2JC1GIzYjRlxxRjtGK0YrRistRiM2JUYrRmVwRisvJS5saW5ldGhpY2tuZXNzR0ZccC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zdcy8lKWJldmVsbGVkR0ZDRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGZ3MvJSpsaW5lYnJlYWtHUSVhdXRvRictRjU2L1EwJkRpZmZlcmVudGlhbEQ7RidGOEY7Rj5GQUZERkZGSEZKRkxGTkZQRlNGXHFGKw== tamb\303\251m n\303\243o existe.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Quest\303\243o 1</Font></Text-field> Encontrar as integrais impr\303\263pria a seguir. Caso n\303\243o seja poss\303\255vel, mostre que n\303\243o existem. (N\303\243o \303\251 aconselh\303\241vel calcular com um \303\272nico passo do Maple: calcule-as como limites de integrais pr\303\263prias.) Int((2*x + 1)/(x^4 + 1), x=0..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JComLCZJInhHRigiIiMiIiJGM0YzLCYqJEYxIiIlRjNGM0YzISIiL0YxOyIiIUkpaW5maW5pdHlHRiZGKw== inttob := int((2*x + 1)/(x^4 + 1), x=0..b); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2JHRigsKiomIiIjIyIiIkYwLUknYXJjdGFuR0YlNiMsJiomSSJiR0YoRjJGMEYxRjJGMkYyRjIjRjIiIiUqJkYwRjEtRjQ2IywmRjdGMiEiIkYyRjJGOSomRjBGMS1JI2xuR0YlNiMsJComLCgqJEY4RjBGMkY3RjJGMkYyRjIsKEY3RjJGR0Y/Rj9GMkY/Rj9GMiNGMiIiKS1GNDYjRkdGMjcjRi4= limit(inttob, b=infinity); LCYqJiIiIyMiIiJGJEkjUGlHJSpwcm90ZWN0ZWRHRiYjRiYiIiVGJ0Yl Int(1/(x*ln(x)), x=2..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JComSSJ4R0YoISIiLUkjbG5HRiU2I0YwRjEvRjA7IiIjSSlpbmZpbml0eUdGJkYr inttob := int(1/(x*ln(x)), x=2..b); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2JHRigtSSRpbnRHRiU2JComSSJ4R0YoISIiLUkjbG5HRiU2I0YyRjMvRjI7IiIjSSJiR0YoNyNGLg== limit(inttob, b=infinity); SSlpbmZpbml0eUclKnByb3RlY3RlZEc= (Verifica-se que esta integral impr\303\263pria n\303\243o existe.) Int((2*x + 1)/(x^4 + 1), x=-infinity..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JComLCZJInhHRigiIiMiIiJGM0YzLCYqJEYxIiIlRjNGM0YzISIiL0YxOywkSSlpbmZpbml0eUdGJkY3RjtGKw== part1 := int((2*x + 1)/(x^4 + 1), x=-a..0); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MUdGKCwqKiYiIiMjIiIiRjAtSSdhcmN0YW5HRiU2IywmKiZJImFHRihGMkYwRjFGMiEiIkYyRjIjRjIiIiUqJkYwRjEtRjQ2IywmRjdGMkYyRjJGMkY6KiZGMEYxLUkjbG5HRiU2IywkKiYsKEY3RjIqJEY4RjBGOUY5RjJGMiwoRkdGMkY3RjJGMkYyRjlGOUYyI0Y5IiIpLUY0NiNGR0Y5NyNGLg== limit1 := limit(part1, a=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDFHRigsJiomIiIjIyIiIkYwSSNQaUdGJkYyI0YyIiIlRjMjISIiRjA3I0Yu part2 := int((2*x + 1)/(x^4 + 1), x=0..b); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MkdGKCwqKiYiIiMjIiIiRjAtSSdhcmN0YW5HRiU2IywmKiZJImJHRihGMkYwRjFGMkYyRjJGMiNGMiIiJSomRjBGMS1GNDYjLCZGN0YyISIiRjJGMkY5KiZGMEYxLUkjbG5HRiU2IywkKiYsKCokRjhGMEYyRjdGMkYyRjJGMiwoRjdGMkZHRj9GP0YyRj9GP0YyI0YyIiIpLUY0NiNGR0YyNyNGLg== limit2 := limit(part2, b=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDJHRigsJiomIiIjIyIiIkYwSSNQaUdGJkYyI0YyIiIlRjNGMTcjRi4= Como ambas as partes das integral impr\303\263pria possuem limites, separadamente, deve-se adicion\303\241-las para que se obtenha a resposta final. Int((2*x + 1)/(x^4 + 1), x=-infinity..infinity) = limit1 + limit2; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJiwmSSJ4R0YoIiIjIiIiRjRGNCwmKiRGMiIiJUY0RjRGNCEiIi9GMjssJEkpaW5maW5pdHlHRiZGOEY8LCQqJkYzI0Y0RjNJI1BpR0YmRjRGP0Yr Int(sin(x)/(x^2 + 1), x=-infinity..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JComLUkkc2luR0YlNiNJInhHRigiIiIsJiokRjMiIiNGNEY0RjQhIiIvRjM7LCRJKWluZmluaXR5R0YmRjhGPEYr part1 := int(sin(x)/(x^2 + 1), x=-a..0); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MUdGKCwwKiYtSSNDaUdGJTYjLCZJImFHRighIiJeI0Y1IiIiRjctSSVzaW5oR0YlNiNGN0Y3I0Y1IiIjKiheI0Y7RjctSSNTaUdGJTYjLCZGNEY3XiNGN0Y3RjctSSVjb3NoR0YlRjpGN0Y3KiYtRjE2IywmRjRGNUZDRjdGN0Y4RjdGOyooXiMjRjdGPEY3LUZANiMsJkY0RjdGNkY3RjdGREY3RjcqJi1GMTYjRkNGN0Y4RjdGNyooRj5GN0kjUGlHRiZGN0Y4RjdGNyooRkNGNy1GQEZSRjdGREY3Rjc3I0Yu limit1 := limit(part1, a=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDFHRigsKComLUkjQ2lHRiU2I14jIiIiRjQtSSVzaW5oR0YlNiNGNEY0RjQqKF4jIyEiIiIiI0Y0SSNQaUdGJkY0RjVGNEY0KihGM0Y0LUkjU2lHRiVGMkY0LUklY29zaEdGJUY3RjRGNDcjRi4= part2 := int(sin(x)/(x^2 + 1), x=0..b); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MkdGKCwwKiYtSSNDaUdGJTYjXiMiIiJGNC1JJXNpbmhHRiU2I0Y0RjQhIiIqKF4jI0Y0IiIjRjRJI1BpR0YmRjRGNUY0RjQqKF4jRjhGNC1JI1NpR0YlRjJGNC1JJWNvc2hHRiVGN0Y0RjQqJi1GMTYjLCZJImJHRihGNEY/RjRGNEY1RjRGOyooXiMjRjhGPEY0LUZBRkZGNEZCRjRGNComLUYxNiMsJkZIRjRGM0Y0RjRGNUY0RjsqKEY6RjQtRkFGT0Y0RkJGNEY0NyNGLg== limit2 := limit(part2, b=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDJHRigsKComLUkjQ2lHRiU2I14jIiIiRjQtSSVzaW5oR0YlNiNGNEY0ISIiKiheIyNGNCIiI0Y0SSNQaUdGJkY0RjVGNEY0KiheI0Y4RjQtSSNTaUdGJUYyRjQtSSVjb3NoR0YlRjdGNEY0NyNGLg== Estes limites envolvem algumas fun\303\247\303\265es n\303\243o familiares. Por\303\251m, (ambas) existem, e podem ser adicionadas a fim de obter-se a resposta. Int(sin(x)/(x^2 + 1), x=-infinity..infinity) = limit1 + limit2; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJi1JJHNpbkdGJTYjSSJ4R0YoIiIiLCYqJEY0IiIjRjVGNUY1ISIiL0Y0OywkSSlpbmZpbml0eUdGJkY5Rj0iIiFGKw== A resposta final \303\251 menos surpreendente quando compreendemos que a fun\303\247\303\243o integrada \303\251 \303\255mpar, e os limites da integral s\303\243o sim\303\251tricos em torno de 0. Int(x*sin(x)/(x^2 + 1), x=-infinity..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JCooSSJ4R0YoIiIiLUkkc2luR0YlNiNGMEYxLCYqJEYwIiIjRjFGMUYxISIiL0YwOywkSSlpbmZpbml0eUdGJkY4RjxGKw== part1 := int(x*sin(x)/(x^2 + 1), x=-a..0); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MUdGKCwsKiYtSSNTaUdGJTYjLCZJImFHRigiIiJeI0Y1RjVGNS1JJWNvc2hHRiU2I0Y1RjUjRjUiIiMqKF4jIyEiIkY7RjUtSSNDaUdGJTYjLCZGNEY/XiNGP0Y1RjUtSSVzaW5oR0YlRjlGNUY1KiYtRjE2IywmRjRGNUZERjVGNUY3RjVGOiooXiNGOkY1LUZBNiMsJkY0Rj9GNkY1RjVGRUY1RjUqJkkjUGlHRiZGNUZFRjVGOjcjRi4= limit1 := limit(part1, a=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDFHRigsJiomSSNQaUdGJiIiIi1JJXNpbmhHRiU2I0YxRjEjISIiIiIjKiZGMEYxLUklY29zaEdGJUY0RjEjRjFGNzcjRi4= part2 := int(x*sin(x)/(x^2 + 1), x=0..b); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MkdGKCwsKiZJI1BpR0YmIiIiLUklc2luaEdGJTYjRjFGMSMhIiIiIiMqJi1JI1NpR0YlNiMsJkkiYkdGKEYxXiNGNkYxRjEtSSVjb3NoR0YlRjRGMSNGMUY3KiheI0ZBRjEtSSNDaUdGJUY7RjFGMkYxRjEqJi1GOjYjLCZGPUYxXiNGMUYxRjFGP0YxRkEqKF4jRjVGMS1GRUZIRjFGMkYxRjE3I0Yu limit2 := limit(part2, b=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDJHRigsJiomSSNQaUdGJiIiIi1JJXNpbmhHRiU2I0YxRjEjISIiIiIjKiZGMEYxLUklY29zaEdGJUY0RjEjRjFGNzcjRi4= Int(x*sin(x)/(x^2 + 1), x=-infinity..infinity) = limit1 + limit2; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqKEkieEdGKCIiIi1JJHNpbkdGJTYjRjFGMiwmKiRGMSIiI0YyRjJGMiEiIi9GMTssJEkpaW5maW5pdHlHRiZGOUY9LCYqJkkjUGlHRiZGMi1JJXNpbmhHRiU2I0YyRjJGOSomRkBGMi1JJWNvc2hHRiVGQ0YyRjJGKw== Int(2*x/(x^2 + 5), x=-infinity..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JCwkKiZJInhHRigiIiIsJiokRjEiIiNGMiIiJkYyISIiRjUvRjE7LCRJKWluZmluaXR5R0YmRjdGO0Yr part1 := int(2*x/(x^2 + 5), x=-a..0); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MUdGKCwmLUkjbG5HRiU2IywmKiRJImFHRigiIiMiIiIiIiZGNiEiIi1GMDYjRjdGNjcjRi4= limit1 := limit(part1, a=infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdsaW1pdDFHRigsJEkpaW5maW5pdHlHRiYhIiI3I0Yu Como uma das duas partes da integral impr\303\263pria n\303\243o existe, deve-se encerrar os c\303\241lculos: 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 n\303\243o existe.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Tipo 2: Fun\303\247\303\265es Ilimitadas</Font></Text-field> O segundo tipo de integral impr\303\263pria \303\251 aquele em que a fun\303\247\303\243o \303\251 ilimitada no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic= . Um exemplo comum ocorre quando a fun\303\247\303\243o torna-se ilimitada em uma das extremidades do intervalo. Examinaremos este caso. (Como explicado anteriormente, outros casos podem ser reduzidos a este.) Suponhamos inicialmente que, para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRKiZlcHNpbG9uO0YnLyUnaXRhbGljR0Y+RjVGKw== pequeno, f \303\251 integr\303\241vel em todo o intervalo (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] . Portanto, a integral impr\303\263pria \303\251 definida como 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 . Apresentamos um exemplo a seguir. Verifica-se que o Maple tamb\303\251m pode resolver este tipo de integral impr\303\263pria. Int(1/sqrt(x), x=0..2) = int(1/sqrt(x), x=0..2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJEkieEdGKCMhIiIiIiMvRjE7IiIhRjQsJCokRjQjIiIiRjRGNEYr Para verificar a resposta, resolvemos a integral de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEqJmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== at\303\251 2 e fazemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEqJmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== aproximar-se de 0 pela direita. (Note o uso do argumento right no comando limit.) ad := int(1/sqrt(x), x); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhZEdGKCwkKiRJInhHRigjIiIiIiIjRjM3I0Yu properint := subs(x=2, ad) - subs(x=epsilon, ad); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpwcm9wZXJpbnRHRigsJiokIiIjIyIiIkYwRjAqJEkoZXBzaWxvbkdGKEYxISIjNyNGLg== Int(1/sqrt(x), x=0..2) = limit(properint, epsilon=0,right); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJEkieEdGKCMhIiIiIiMvRjE7IiIhRjQsJCokRjQjIiIiRjRGNEYr A prop\303\263sito, note que o limite de dois lados, 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, n\303\243o existe. Por\303\251m, ao menos neste caso, o Maple n\303\243o nos informa: limit(properint, epsilon=0); LCQqJCIiIyMiIiJGJEYk No caso em que f torna-se ilimitado na extremidade direita, define-se 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 . Int(1/(x-2)^2, x=0..2) = int(1/(x-2)^2, x=0..2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJCwmSSJ4R0YoIiIiISIjRjNGNC9GMjsiIiEiIiNJKWluZmluaXR5R0YmRis= A resposta do Maple sugere que a integral impr\303\263pria n\303\243o exite. Verifiquemos: ad := int(1/(x-2)^2, x); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhZEdGKCwkKiQsJkkieEdGKCIiIiEiI0YyISIiRjQ3I0Yu properint := subs(x=2-epsilon, ad) - subs(x=0, ad); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpwcm9wZXJpbnRHRigsJiokSShlcHNpbG9uR0YoISIiIiIiI0YxIiIjRjI3I0Yu Pode-se notar que o limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSomZXBzaWxvbjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEnJnJhcnI7RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIwRidGN0Yr n\303\243o existe. Pelo Maple: limit(properint, epsilon=0, right); SSlpbmZpbml0eUclKnByb3RlY3RlZEc= Finalmente, se f \303\251 ilimitada em ambas as extremidades, deve-se dividir o intervalo em algum ponto conveniente, c, e define-se 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 , contanto que ambas as integrais impr\303\263prias do lado direito existam.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Quest\303\243o 2</Font></Text-field> Resolver as integrais impr\303\263prias a seguir. Caso n\303\243o seja poss\303\255vel, mostrar que n\303\243o existem. Int(1/sqrt(1-x), x=0..1); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JCokLCYiIiJGMUkieEdGKCEiIiNGMyIiIy9GMjsiIiFGMUYr inttoe := int(1/sqrt(1-x), x=0..1-epsilon); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2VHRigsJiIiIyIiIiokSShlcHNpbG9uR0YoI0YwRi8hIiM3I0Yu Int(1/sqrt(1-x), x=0..1) = limit(inttoe, epsilon=0, right); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJCwmIiIiRjJJInhHRighIiIjRjQiIiMvRjM7IiIhRjJGNkYr Int(1/sqrt(4-x^2), x=0..2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JCokLCYqJEkieEdGKCIiIyEiIiIiJSIiIiNGNEYzL0YyOyIiIUYzRis= inttoe := int(1/sqrt(4-x^2), x=0..2-epsilon); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2VHRigsJC1JJ2FyY3NpbkdGJTYjLCYhIiIiIiJJKGVwc2lsb25HRigjRjQiIiNGMzcjRi4= Int(1/sqrt(4-x^2), x=0..2) = limit(inttoe, epsilon=0, right); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJCwmKiRJInhHRigiIiMhIiIiIiUiIiIjRjVGNC9GMzsiIiFGNCwkSSNQaUdGJiNGN0Y0Ris= Int(1/(x^2 - 1), x=1..2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JCokLCYqJEkieEdGKCIiIyIiIiEiIkY0RjUvRjI7RjRGM0Yr inttoe := int(1/(x^2 - 1), x=1+epsilon..2); Warning, unable to determine if -1 is between 1+epsilon and 2; try to use assumptions or use the AllSolutions option Warning, unable to determine if 1 is between 1+epsilon and 2; try to use assumptions or use the AllSolutions option LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2VHRigtSSRpbnRHRiU2JCokLCYqJEkieEdGKCIiIyIiIiEiIkY2RjcvRjQ7LCZGNkY2SShlcHNpbG9uR0YoRjZGNTcjRi4= limit(inttoe, epsilon=0, right); Warning, unable to determine if -1 is between 1+epsilon and 2; try to use assumptions or use the AllSolutions option Warning, unable to determine if 1 is between 1+epsilon and 2; try to use assumptions or use the AllSolutions option SSlpbmZpbml0eUclKnByb3RlY3RlZEc= Esta integral impr\303\263pria n\303\243o existe. Int(exp(sqrt(x))/sqrt(x), x=0..1); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JComLUkkZXhwR0YlNiMqJEkieEdGKCMiIiIiIiNGNkY0IyEiIkY3L0Y0OyIiIUY2Ris= inttoe := int(exp(sqrt(x))/sqrt(x), x=epsilon..1); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2VHRigsJi1JJGV4cEdGJTYjKiRJKGVwc2lsb25HRigjIiIiIiIjISIjLUYwNiNGNUY2NyNGLg== Int(exp(sqrt(x))/sqrt(x), x=0..1) = limit(inttoe, epsilon=0, right); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJi1JJGV4cEdGJTYjKiRJInhHRigjIiIiIiIjRjdGNSMhIiJGOC9GNTsiIiFGNywmISIjRjctRjI2I0Y3RjhGKw== Int(1/(x^2 - 5*x + 6), x=1..4); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRiU2JCokLCgqJEkieEdGKCIiIyIiIkYyISImIiInRjQhIiIvRjI7RjQiIiVGKw== Note que o denominador desta fun\303\247\303\243o \303\251 igual a 0 em dois lugares: x = 2 e x = 3. Portanto, deve-se dividir a integral em (pelo menos) 3 partes, resolv\303\252-las separadamente e ent\303\243o adicion\303\241-las. Primeiramente, deve-se tratar a integral de 1 a 2, sendo esta impr\303\263pria em 2. inttoe := int(1/(x^2 - 5*x + 6), x=1..2-epsilon); Warning, unable to determine if 2 is between 1 and 2-epsilon; try to use assumptions or use the AllSolutions option Warning, unable to determine if 3 is between 1 and 2-epsilon; try to use assumptions or use the AllSolutions option LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbnR0b2VHRigtSSRpbnRHRiU2JCokLCgqJEkieEdGKCIiIyIiIkY0ISImIiInRjYhIiIvRjQ7RjYsJkY1RjZJKGVwc2lsb25HRihGOTcjRi4= part1 := limit(inttoe, epsilon=0, right); Warning, unable to determine if 2 is between 1 and 2-epsilon; try to use assumptions or use the AllSolutions option Warning, unable to determine if 3 is between 1 and 2-epsilon; try to use assumptions or use the AllSolutions option LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MUdGKEkpaW5maW5pdHlHRiY3I0Yu Deve-se parar neste momento: a primeira parte da integral n\303\243o existe e, portanto, a integral completa tamb\303\251m n\303\243o existe.