C\303\241lculo II

Li\303\247\303\243o 9: Integra\303\247\303\243o por Substitui\303\247\303\243o: Exemplos

Apresentamos a seguir 20 anti-derivadas. Cada uma delas \303\251 reduzida a uma forma mais simples por meio de substitui\303\247\303\243o. Seguem algumas regras:

Utilizar o comando n\303\243o avaliado Int.

O problema \303\251 considerado "resolvido" quando for reduzido a um problema de forma conhecida, ou que possa ser resolvido por uma f\303\263rmula de redu\303\247\303\243o conhecida, ou alguma soma de problemas destes tipos.

Encontrar a substitui\303\247\303\243o mais simples para cada caso.

restart;

\303\211 preciso carregar o pacote student, caso contr\303\241rio o comando changevar n\303\243o poder\303\241 ser usado.

with(student);

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1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1JJm1mcmFjR0YkNigtRiM2Iy1JJm1zcXJ0R0YkNiMtRiM2Ji1JI21uR0YkNiRRIjlGJ0Y4LUYyNi1RKCZtaW51cztGJ0Y4Rj5GQUZDRkVGR0ZJRksvRk5RLDAuMjIyMjIyMmVtRicvRlFGXm8tRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnLUZnbjYkUSIyRidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKy1GIzYlRitGYG9GKy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGanAvJSliZXZlbGxlZEdGQEYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmRxLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYyNi9RMCZEaWZmZXJlbnRpYWxEO0YnRjVGOEY7Rj5GQUZDRkVGR0ZJRktGTUZQRmVvRis=

p1 := Int(sqrt(9 - x^2)/x^2, x);

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

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Infelizmente, o comando simplify n\303\243o apresenta o resultado desejado neste caso:

simplify(p2);

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A raz\303\243o para isto \303\251 que o Maple n\303\243o simplifica LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbXNxcnRHRiQ2Iy1GIzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn em a, a menos que a seja positivo. \303\211 preciso assumir cos(u) positivo para que o comando simplify funcione. (Por que \303\251 poss\303\255vel fazer tal suposi\303\247\303\243o?)

assume(cos(u) >= 0);

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKC1JJEludEdGJzYkKiYtSSRjb3NHRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIjLUkkc2luR0YlRjQhIiNGNTcjRi4=

p4 := subs(-1 + cos(u)^2 = -sin(u)^2, p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKC1JJEludEdGJzYkKiYtSSRjb3NHRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIjLUkkc2luR0YlRjQhIiNGNTcjRi4=

p5 := subs(cos(u)^2 / sin(u)^2 = cot(u)^2, p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKC1JJEludEdGJzYkKiQtSSRjb3RHRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIjRjU3I0Yu

p6 := subs(cot(u)^2 = csc(u)^2 - 1, p5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNkdGKC1JJEludEdGJzYkLCYqJC1JJGNzY0dGJTYjSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigiIiMiIiIhIiJGPEY2NyNGLg==

Agora basta resolver LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GIzYjLUklbXN1cEdGJDYlLUYjNiUtRiw2JVEkY3NjRicvJSdpdGFsaWNHRkBGOC1GMjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y4Rj5GQUZDRkVGR0ZJRktGTUZQLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEidUYnL0ZnblEldHJ1ZUYnL0Y5USdpdGFsaWNGJ0Y4LUkjbW5HRiQ2JFEiMkYnRjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGY3AvJSpsaW5lYnJlYWtHUSVhdXRvRictRjI2L1EwJkRpZmZlcmVudGlhbEQ7RidGNUY4RjtGPkZBRkNGRUZHRklGS0ZNRlBGYG9GKw== e 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.

2. 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

p1 := Int(1/(x^2 * sqrt(x^2 + 4)), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRighIiMsJiokRjIiIiMiIiIiIiVGNyMhIiJGNkYyNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqKCwmIiIiRjQqJC1JJHRhbkdGJTYjSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigiIiNGNEY0RjYhIiMsJkY1IiIlRkFGNCMhIiJGPiNGNEY+Rjk3I0Yu

p3 := simplify(p2);

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3. 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

p1 := Int(x^2 / sqrt(x^2 + 4), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiMsJiokRjJGMyIiIiIiJUY2IyEiIkYzRjI3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqKC1JJHRhbkdGJTYjSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigiIiMsJiIiIkY9KiRGM0Y7Rj1GPSwmRj4iIiVGQEY9IyEiIkY7IiIpRjY3I0Yu

p3 := simplify(p2);

4. 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

p1 := Int(x / sqrt(x^2 + 4), x);

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p2 := changevar(u=x^2 + 4, p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJEkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIyEiIiIiIyMiIiJGOkYzNyNGLg==

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJEkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIyEiIiIiI0YzIyIiIkY6NyNGLg==

5. 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

p1 := Int(1/sqrt(x^2 - 4), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQsJiokSSJ4R0YoIiIjIiIiISIlRjYjISIiRjVGNDcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqKCwmKiQtSSRzZWNHRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIjIiIlISIlIiIiIyEiIkY9RjVGQC1JJHRhbkdGJUY3RkBGPUY4NyNGLg==

p3 := simplify(p2);

6. 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

p1 := Int( sec(x)*tan(x)*sqrt(1 + sec(x)), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKigtSSRzZWNHRiU2I0kieEdGKCIiIi1JJHRhbkdGJUY0RjYsJkY2RjZGMkY2I0Y2IiIjRjU3I0Yu

p2 := changevar(u = 1 + sec(x), p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiRJI3V8aXJHNiMvSSdfbG9jYWxHRiZJMF9uMTMzNzIwMTk1NTIwMEdGKCMiIiIiIiNGMjcjRi4=

7. 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

p1 := Int(x*sqrt(3*x + 1), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiIsJkYyIiIkRjNGMyNGMyIiI0YyNyNGLg==

p2 := changevar(u = 3*x+ 1, p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJiwmSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigjIiIiIiIkIyEiIkY7RjpGOkY0I0Y6IiIjRjlGNDcjRi4=

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJiwmSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigiIiIhIiJGOUY5RjQjRjkiIiNGNCNGOSIiKjcjRi4=

8. 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

p1 := Int(x^3 * sqrt(9 - x^2), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiQsJiokRjIiIiMhIiIiIioiIiIjRjlGNkYyNyNGLg==

p2 := changevar(u = 9 - x^2, p1, u);

p3 := simplify(p2);

9. 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

p1 := Int(cos(x)^4 * sin(x), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSRjb3NHRiU2I0kieEdGKCIiJS1JJHNpbkdGJUY0IiIiRjU3I0Yu

p2 := changevar(u = cos(x), p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJEkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIlISIiRjM3I0Yu

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJEkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIlRjMhIiI3I0Yu

10. 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

p1 := Int(1/( (x+1)^2 + 1)^2, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQsJiokLCZJInhHRigiIiJGNkY2IiIjRjZGNkY2ISIjRjU3I0Yu

p2 := changevar(x+1 = tan(u), p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiQsJiIiIkYzKiQtSSR0YW5HRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIiIjRjMhIiJGODcjRi4=

11. 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

p1 := Int(x^3 * sqrt(4 - 9*x^2), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiQsJiokRjIiIiMhIioiIiUiIiIjRjlGNkYyNyNGLg==

p2 := changevar(u = 4 - 9*x^2, p1, u);

p3 := simplify(p2);

12. 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

p1 := Int(x*sqrt(1 + x^4), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiIsJiokRjIiIiVGM0YzRjMjRjMiIiNGMjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJCwmKiRJI3V8aXJHNiMvSSdfbG9jYWxHRiZJMF9uMTMzNzIwMTk1NTIwMEdGKCIiIyIiIkY7RjsjRjtGOkY8RjU3I0Yu

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJCwmKiRJI3V8aXJHNiMvSSdfbG9jYWxHRiZJMF9uMTMzNzIwMTk1NTIwMEdGKCIiIyIiIkY7RjsjRjtGOkY1Rjw3I0Yu

p4 := changevar(u = tan(t), p3, t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkLUkkSW50R0YnNiQqJCwmIiIiRjQqJC1JJHRhbkdGJTYjSSJ0R0YoIiIjRjQjIiIkRjpGOSNGNEY6NyNGLg==

p5 := subs(1 + tan(t)^2 = sec(t)^2, p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkLUkkSW50R0YnNiQqJCokLUkkc2VjR0YlNiNJInRHRigiIiMjIiIkRjhGNyMiIiJGODcjRi4=

assume(sec(t) > 0);

p6 := simplify(p5, power);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNkdGKCwkLUkkSW50R0YnNiQqJC1JJHNlY0dGJTYjSSN0fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcxOTA3MDI1MjhHRigiIiRGNiMiIiIiIiM3I0Yu

13. 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

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiYsJiokRjIiIiQiIiJGN0Y3I0Y3RjZGMjcjRi4=

p2 := changevar(u = x^3 + 1, p1, u);

p3 := simplify(p2);

14. 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

p1 := Int(x/sqrt(1 + 2*x), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiIsJkYzRjNGMiIiIyMhIiJGNUYyNyNGLg==

p2 := changevar(u = 1 + 2*x, p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJiwmIyEiIiIiIyIiIkkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoI0Y3RjZGN0Y4RjRGPUY4NyNGLg==

p3 := simplify(p2);

15. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GIzYlLUklbXN1cEdGJDYlLUYyNi1RLyZFeHBvbmVudGlhbEU7RidGOEY+RkFGQ0ZFRkdGSUZLRk0vRlFRLDAuMTExMTExMWVtRictRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjI2LlExJkludmlzaWJsZVRpbWVzO0YnL0Y2UShbMCwwLDBdRidGOEY+RkFGQ0ZFRkdGSUZLRk1GUC1JJm1zcXJ0R0YkNiMtRiM2JS1JI21uR0YkNiRRIjlGJ0Y4LUYyNi1RKCZtaW51cztGJ0Y4Rj5GQUZDRkVGR0ZJRksvRk5RLDAuMjIyMjIyMmVtRicvRlFGY3AtRlU2JUZXLUYjNiUtRlxwNiRRIjJGJ0Y4RmFvRmZuRl5vRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGYXEvJSpsaW5lYnJlYWtHUSVhdXRvRictRjI2L1EwJkRpZmZlcmVudGlhbEQ7RidGNUY4RjtGPkZBRkNGRUZHRklGS0ZNRlBGZm5GKw==

p1 := Int( exp(x)*sqrt(9 - exp(2*x)), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSRleHBHRiU2I0kieEdGKCIiIiwmIiIqRjYtRjM2IywkRjUiIiMhIiIjRjZGPEY1NyNGLg==

p2 := changevar(u = exp(x), p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiQsJiokSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigiIiMhIiIiIioiIiIjRjxGOUY0NyNGLg==

p3 := changevar(u = 3*sin(theta), p2, theta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKC1JJEludEdGJzYkLCQqJiwmKiQtSSRzaW5HRiU2I0kmdGhldGFHRigiIiMhIioiIioiIiIjRjxGOS1JJGNvc0dGJUY3RjwiIiRGODcjRi4=

assume(cos(theta) >= 0);

p4 := simplify(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSd0aGV0YXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjM1NDE2NDQ4R0YoIiIjRjYiIio3I0Yu

16. 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

p1 := Int(arctan(x)/(1 + x^2), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSdhcmN0YW5HRiU2I0kieEdGKCIiIiwmKiRGNSIiI0Y2RjZGNiEiIkY1NyNGLg==

p2 := changevar(u = arctan(x), p1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRihGMTcjRi4=

17. 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

p1 := Int(x^4 / sqrt(x^10 - 4), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiUsJiokRjIiIzUiIiIhIiVGNyMhIiIiIiNGMjcjRi4=

p2 := changevar(u = x^5, p1, u);

p3 := simplify(p2);

p4 := changevar(u = 2*sec(phi), p3, phi);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkLUkkSW50R0YnNiQsJCooLCYqJC1JJHNlY0dGJTYjSSRwaGlHRigiIiMiIiUhIiUiIiIjISIiRjpGNkY9LUkkdGFuR0YlRjhGPUY6RjkjRj0iIiY3I0Yu

assume(tan(phi) > 0);

p5 := simplify(p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkLUkkSW50R0YnNiQqKComLUkkc2luR0YlNiNJJXBoaXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjMwMDk5MTY4R0YoIiIjLUkkY29zR0YlRjYhIiMjISIiRjxGPUY/RjQiIiJGNyNGQiIiJjcjRi4=

p6 := subs(1/cos(phi) = sec(phi), p5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNkdGKCwkLUkkSW50R0YnNiQqKComLUkkc2luR0YlNiNJJXBoaXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjMwMDk5MTY4R0YoIiIjLUkkY29zR0YlRjYhIiMjISIiRjxGPUY/RjQiIiJGNyNGQiIiJjcjRi4=

18. 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

(Este problema e o dois pr\303\263ximos est\303\243o relacionados; utilizaremos novas vari\303\241veis para que possamos referirmo-nos a esta resposta sem confus\303\243o.)

q1 := Int((x+2)/sqrt(x^2 + 4*x + 6), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiYsJkkieEdGKCIiIiIiI0Y0RjQsKCokRjNGNUY0RjMiIiUiIidGNCMhIiJGNUYzNyNGLg==

q2 := changevar(u = x^2 + 4*x + 6, q1, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKC1JJEludEdGJzYkLCQqJEkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIyEiIiIiIyMiIiJGOkYzNyNGLg==

q3 := simplify(q2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwkLUkkSW50R0YnNiQqJEkjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoIyEiIiIiI0YzIyIiIkY6NyNGLg==

q4 := value(q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCokSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRigjIiIiIiIjNyNGLg==

q5 := subs(u = x^2 + 4*x + 6, q4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKCokLCgqJEkieEdGKCIiIyIiIkYxIiIlIiInRjMjRjNGMjcjRi4=

q1 = q5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiwmSSJ4R0YoIiIiIiIjRjNGMywoKiRGMkY0RjNGMiIiJSIiJ0YzIyEiIkY0RjIsJiokRjUjRjNGNEYzSSJDR0YoRjNGKw==

19. 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

r1 := Int(2/sqrt(x^2 + 4*x + 6), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMUdGKC1JJEludEdGJzYkLCQqJCwoKiRJInhHRigiIiMiIiJGNSIiJSIiJ0Y3IyEiIkY2RjZGNTcjRi4=

r2 := simplify(r1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMkdGKCwkLUkkSW50R0YnNiQqJCwoKiRJInhHRigiIiMiIiJGNSIiJSIiJ0Y3IyEiIkY2RjVGNjcjRi4=

cs := completesquare(x^2 + 4*x + 6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjc0dGKCwmKiQsJkkieEdGKCIiIiIiI0YyRjNGMkYzRjI3I0Yu

r3 := subs(x^2 + 4*x + 6 = cs, r2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyM0dGKCwkLUkkSW50R0YnNiQqJCwmKiQsJkkieEdGKCIiIiIiI0Y3RjhGN0Y4RjcjISIiRjhGNkY4NyNGLg==

r4 := changevar(x+2 = sqrt(2)*tan(u), r3, u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNEdGKCwkLUkkSW50R0YnNiQqKCwmIiIjIiIiKiQtSSR0YW5HRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoRjRGNCMhIiJGNEY0I0Y1RjQsJkY1RjVGNkY1RjVGOkY0NyNGLg==

r5 := simplify(r4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNUdGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRighIiJGNiIiIzcjRi4=

assume(sec(u) > 0);

r6 := subs(1 + tan(u)^2 = sec(u)^2, r5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNkdGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRighIiJGNiIiIzcjRi4=

r7 := simplify(r6, power);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyN0dGKCwkLUkkSW50R0YnNiQqJC1JJGNvc0dGJTYjSSN1fGlyRzYjL0knX2xvY2FsR0YmSTBfbjEzMzcyMDE5NTUyMDBHRighIiJGNiIiIzcjRi4=

r8 := value(r7);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyOEdGKCwkLUkjbG5HRiU2IywmLUkkc2VjR0YlNiNJI3V8aXJHNiMvSSdfbG9jYWxHRiZJMF9uMTMzNzIwMTk1NTIwMEdGKCIiIi1JJHRhbkdGJUY1RjsiIiM3I0Yu

r9 := subs(tan(u) = (x+2)/sqrt(2), sec(u) = sqrt((x^2 + 4*x + 6)/2), r8);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyOUdGKCwkLUkjbG5HRiU2IywmLUkkc2VjR0YlNiNJI3V8aXJHNiMvSSdfbG9jYWxHRiZJMF9uMTMzNzIwMTk1NTIwMEdGKCIiIi1JJHRhbkdGJUY1RjsiIiM3I0Yu

r10 := simplify(r9);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyMTBHRigsJi1JI2xuR0YlNiMtSSRjb3NHRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoISIjLUYwNiMsJiIiIkY+LUkkc2luR0YlRjRGPiIiIzcjRi4=

r1 = r10 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQsJCokLCgqJEkieEdGKCIiIyIiIkY0IiIlIiInRjYjISIiRjVGNUY0LCgtSSNsbkdGJTYjLUkkY29zR0YlNiNJI3V8aXJHNiMvSSdfbG9jYWxHRiZJMF9uMTMzNzIwMTk1NTIwMEdGKCEiIy1GPTYjLCZGNkY2LUkkc2luR0YlRkFGNkY1SSJDR0YoRjZGKw==

20. 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

s1 := Int(x / sqrt(x^2 + 4*x + 6), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNzMUdGKC1JJEludEdGJzYkKiZJInhHRigiIiIsKCokRjIiIiNGM0YyIiIlIiInRjMjISIiRjZGMjcjRi4=

Observe que esta anti-derivada \303\251 simplesmente a diferen\303\247a das duas anteriores:

s1 = q5 - r10 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJkkieEdGKCIiIiwoKiRGMSIiI0YyRjEiIiUiIidGMiMhIiJGNUYxLCoqJEYzI0YyRjVGMi1JI2xuR0YlNiMtSSRjb3NHRiU2I0kjdXxpckc2Iy9JJ19sb2NhbEdGJkkwX24xMzM3MjAxOTU1MjAwR0YoRjUtRj42IywmRjJGMi1JJHNpbkdGJUZCRjIhIiNJIkNHRihGMkYr