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

C\303\241lculo II

Li\303\247\303\243o 12: Integra\303\247\303\243o de Pot\303\252ncias Trigonom\303\251tricas

restart;

Uma das mais importantes aplica\303\247\303\265es das f\303\263rmulas de redu\303\247\303\243o encontra-se na resolu\303\247\303\243o de anti-derivadas de pot\303\252ncias de fun\303\247\303\265es trigonom\303\251tricas, tais como 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, 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, ou mesmo 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 . Nesta folha de trabalho estudaremos como faz\303\252-lo.

J\303\241 estudados algumas integrais mais simples, como:

Int(sin(x),x) = int(sin(x), x) + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQtSSRzaW5HRiU2I0kieEdGKEYzLCYtSSRjb3NHRiVGMiEiIkkiQ0dGKCIiIkYr

Int(cos(x), x) = int(cos(x), x) + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQtSSRjb3NHRiU2I0kieEdGKEYzLCYtSSRzaW5HRiVGMiIiIkkiQ0dGKEY3Ris=

Int(sec(x)^2,x) = int(sec(x)^2, x) + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YlNiQqJC1JJHNlY0dGJTYjSSJ4R0YoIiIjRjQsJi1JJHRhbkdGJUYzIiIiSSJDR0YoRjlGKw==

(N\303\243o \303\251 simples fazer com que o Maple escreva tan( x ) na \303\272ltima express\303\243o -- pois este prefere trabalhar com senos e cosenos.)

Algumas outras integrais que envolvem pequenas pot\303\252ncias podem ser rapidamente resolvidas por meio de substitui\303\247\303\243o:

with(student):

p1 := Int(sin(x)^5 * cos(x), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSRzaW5HRiU2I0kieEdGKCIiJi1JJGNvc0dGJUY0IiIiRjU3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiRJInVHRigiIiZGMjcjRi4=

p3 := value(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkKiRJInVHRigiIicjIiIiRjE3I0Yu

p4 := subs(u=sin(x), p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiQtSSRzaW5HRiU2I0kieEdGKCIiJyMiIiJGNDcjRi4=

p1 = p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHNpbkdGJTYjSSJ4R0YoIiImLUkkY29zR0YlRjMiIiJGNCwmKiRGMSIiJyNGOEY7SSJDR0YoRjhGKw==

(Naturalmente, a resposta pode ser verificada por diferencia\303\247\303\243o.) Integrais da forma 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 podem ser resolvidas de maneira similar, pela substitui\303\247\303\243o de 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. Por outro lado, como a derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR0YW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Yr \303\251 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, \303\251 preciso que se tenha uma integra da forma 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 para que uma substitui\303\247\303\243o envolvendo uma tangente funcione.

Como j\303\241 aprendemos a resolver anti-derivadas de pequenas pot\303\252ncias de algumas fun\303\247\303\265es trigonom\303\251tricas, pode-se analisar alguns exemplos envolvendo pot\303\252ncias maiores.

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

Encontar as anti-derivadas a seguir atrav\303\251s de substitui\303\247\303\265es apropriadas.

Solu\303\247\303\265es. (Certifique-se de ter carregado o pacote student antes de resolver os problemas.)

(a) 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

p1 := Int( cos(x)^3 * sin(x), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSRjb3NHRiU2I0kieEdGKCIiJC1JJHNpbkdGJUY0IiIiRjU3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJEkidUdGKCIiJCEiIkYzNyNGLg==

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJEkidUdGKCIiJEYzISIiNyNGLg==

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiRJInVHRigiIiUjISIiRjE3I0Yu

p5 := subs(u=cos(x), p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkKiQtSSRjb3NHRiU2I0kieEdGKCIiJSMhIiJGNDcjRi4=

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJGNvc0dGJTYjSSJ4R0YoIiIkLUkkc2luR0YlRjMiIiJGNCwmKiRGMSIiJSMhIiJGO0kiQ0dGKEY4Ris=

(b) 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

p1 := Int( cos(x)^9 * sin(x), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSRjb3NHRiU2I0kieEdGKCIiKi1JJHNpbkdGJUY0IiIiRjU3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQqJEkidUdGKCIiKiEiIkYzNyNGLg==

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQqJEkidUdGKCIiKkYzISIiNyNGLg==

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiRJInVHRigiIzUjISIiRjE3I0Yu

p5 := subs(u=cos(x), p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkKiQtSSRjb3NHRiU2I0kieEdGKCIjNSMhIiJGNDcjRi4=

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJGNvc0dGJTYjSSJ4R0YoIiIqLUkkc2luR0YlRjMiIiJGNCwmKiRGMSIjNSMhIiJGO0kiQ0dGKEY4Ris=

(c) 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

p1 := Int( tan(x)^4 * sec(x)^2 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSR0YW5HRiU2I0kieEdGKCIiJS1JJHNlY0dGJUY0IiIjRjU3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiRJInVHRigiIiVGMjcjRi4=

p3 := value(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkKiRJInVHRigiIiYjIiIiRjE3I0Yu

p4 := subs( u=tan(x), p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiQtSSR0YW5HRiU2I0kieEdGKCIiJiMiIiJGNDcjRi4=

p1 = p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHRhbkdGJTYjSSJ4R0YoIiIlLUkkc2VjR0YlRjMiIiNGNCwmKiRGMSIiJiMiIiJGO0kiQ0dGKEY9Ris=

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

Encontrar cada uma das anti-derivadas a seguir por meio de substitui\303\247\303\265es apropriadas.

Solu\303\247\303\265es.

(a) 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

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYpLUkkc2luR0YlNiNJInhHRihJIm5HRigiIiItSSRjb3NHRiVGNUY4RjY3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKUkidUdGKEkibkdGKEYyNyNGLg==

p3 := value(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKComKUkidUdGKCwmSSJuR0YoIiIiRjNGM0YzRjEhIiI3I0Yu

p4 := subs( u=sin(x), p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKComKS1JJHNpbkdGJTYjSSJ4R0YoLCZJIm5HRigiIiJGNkY2RjZGNCEiIjcjRi4=

p1 = p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiktSSRzaW5HRiU2I0kieEdGKEkibkdGKCIiIi1JJGNvc0dGJUY0RjdGNSwmKiYpRjIsJkY2RjdGN0Y3RjdGPSEiIkY3SSJDR0YoRjdGKw==

(b) 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

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYpLUkkY29zR0YlNiNJInhHRihJIm5HRigiIiItSSRzaW5HRiVGNUY4RjY3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQpSSJ1R0YoSSJuR0YoISIiRjM3I0Yu

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQpSSJ1R0YoSSJuR0YoRjMhIiI3I0Yu

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiYpSSJ1R0YoLCZJIm5HRigiIiJGNEY0RjRGMiEiIkY1NyNGLg==

p5 := subs( u=cos(x), p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkKiYpLUkkY29zR0YlNiNJInhHRigsJkkibkdGKCIiIkY3RjdGN0Y1ISIiRjg3I0Yu

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiktSSRjb3NHRiU2I0kieEdGKEkibkdGKCIiIi1JJHNpbkdGJUY0RjdGNSwmKiYpRjIsJkY2RjdGN0Y3RjdGPSEiIkY+SSJDR0YoRjdGKw==

(c) 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

p1 := Int( tan(x)^n * sec(x)^2 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYpLUkkdGFuR0YlNiNJInhHRihJIm5HRigiIiItSSRzZWNHRiVGNSIiI0Y2NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKUkidUdGKEkibkdGKEYyNyNGLg==

p3 := value(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKComKUkidUdGKCwmSSJuR0YoIiIiRjNGM0YzRjEhIiI3I0Yu

p4 := subs( u=tan(x), p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKComKS1JJHRhbkdGJTYjSSJ4R0YoLCZJIm5HRigiIiJGNkY2RjZGNCEiIjcjRi4=

p1 = p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiktSSR0YW5HRiU2I0kieEdGKEkibkdGKCIiIi1JJHNlY0dGJUY0IiIjRjUsJiomKUYyLCZGNkY3RjdGN0Y3Rj4hIiJGN0kiQ0dGKEY3Ris=

(d) 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

p1 := Int( cot(x)^n * csc(x)^2 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYpLUkkY290R0YlNiNJInhHRihJIm5HRigiIiItSSRjc2NHRiVGNSIiI0Y2NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkLCQpSSJ1R0YoSSJuR0YoISIiRjM3I0Yu

p3 := simplify(p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkLUkkSW50R0YnNiQpSSJ1R0YoSSJuR0YoRjMhIiI3I0Yu

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwkKiYpSSJ1R0YoLCZJIm5HRigiIiJGNEY0RjRGMiEiIkY1NyNGLg==

p5 := subs( u=cot(x), p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwkKiYpLUkkY290R0YlNiNJInhHRigsJkkibkdGKCIiIkY3RjdGN0Y1ISIiRjg3I0Yu

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJiktSSRjb3RHRiU2I0kieEdGKEkibkdGKCIiIi1JJGNzY0dGJUY0IiIjRjUsJiomKUYyLCZGNkY3RjdGN0Y3Rj4hIiJGP0kiQ0dGKEY3Ris=

<Text-field style="Heading 1" layout="Heading 1">Integrais da forma <Equation executable="false" style="2D Comment" input-equation="" display="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">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</Equation></Text-field>

restart;

with(student):

Iniciaremos com um caso especial: m = 0. Tem-se, portanto, a integral

p1 := Int(sin(x)^n , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKS1JJHNpbkdGJTYjSSJ4R0YoSSJuR0YoRjU3I0Yu

N\303\243o h\303\241 maiores dificuldades quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw== ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw==. Sendo assim assumimos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGKw==. Neste caso, a f\303\263rmula de redu\303\247\303\243o \303\251 dada por

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 = 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 + 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 .

<Text-field style="Heading 2" layout="Heading 2">Deriva\303\247\303\243o da f\303\263rmula (opcional)</Text-field>

O truque b\303\241sico consiste em isolar um fator 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, e escrev\303\252-lo como 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:

p2 := Int( sin(x)^2*sin(x)^(n-2), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKC1JJEludEdGJzYkKiYtSSRzaW5HRiU2I0kieEdGKCIiIylGMiwmSSJuR0YoIiIiISIjRjpGOkY1NyNGLg==

p3 := subs(sin(x)^2 = (1-cos(x)^2), p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKC1JJEludEdGJzYkKiYsJiIiIkYzKiQtSSRjb3NHRiU2I0kieEdGKCIiIyEiIkYzKS1JJHNpbkdGJUY3LCZJIm5HRihGMyEiI0YzRjNGODcjRi4=

p4 := expand(p3, n-2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwmLUkkSW50R0YnNiQpLUkkc2luR0YlNiNJInhHRigsJkkibkdGKCIiIiEiI0Y5RjZGOS1GMDYkKiZGMkY5LUkkY29zR0YlRjUiIiNGNiEiIjcjRi4=

(O argumento opcional LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GKw== no comando expand previne a expans\303\243o do expoente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GKw==.) Deseja-se agora integrar a segunda integral por partes. Para isso, escrevem\303\264-la como 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, e diferenciamos o segundo fator de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Yr. Isto requer a anti-diferencia\303\247\303\243o do fator 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, o que j\303\241 foi estudado anteriormente. Ao utilizar o Maple neste c\303\241lculo, \303\251 conveniente que se use o comando op. O comando op refere-se ao operando. Este refere-se a um operando particular, ou peda\303\247o deste, em uma express\303\243o. Por exemplo, na express\303\243o p4 existem duas integrais: dois operandos. (Cada operando \303\251 constitu\303\255do de peda\303\247os menores, o que n\303\243o ser\303\241 levado em conta neste momento.) Cada uma das integrais pode ser referenciada pela sele\303\247\303\243o do operando correto de p4:

op(1, p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRJbnRHRic2JCktSSRzaW5HRiU2I0kieEdGKCwmSSJuR0YoIiIiISIjRjZGM0Yr

op(2, p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJC1JJEludEdGJzYkKiYpLUkkc2luR0YlNiNJInhHRigsJkkibkdGKCIiIiEiI0Y4RjgtSSRjb3NHRiVGNCIiI0Y1ISIiRis=

Com o comando op, pode-se integrar apenas uma das duas integrais por partes:

p5 := op(1, p4) + intparts(op(2, p4), cos(x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwoLUkkSW50R0YnNiQpLUkkc2luR0YlNiNJInhHRigsJkkibkdGKCIiIiEiI0Y5RjZGOSooLUkkY29zR0YlRjVGOSlGMywmRjhGOSEiIkY5RjlGP0ZARkAtRjA2JCwkKihGM0Y5Rj5GOUY/RkBGQEY2Rjk3I0Yu

p6 := op(1, p5) + op(2, p5) + simplify(op(3, p5));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNkdGKCwoLUkkSW50R0YnNiQpLUkkc2luR0YlNiNJInhHRigsJkkibkdGKCIiIiEiI0Y5RjZGOSooLUkkY29zR0YlRjVGOSlGMywmRjhGOSEiIkY5RjlGP0ZARkAqJkY/RkAtRjA2JClGM0Y4RjZGOUZANyNGLg==

O problema est\303\241 quase resolvido. A \303\272ltima express\303\243o, p6, \303\251 igual a integral original, p1. O primeiro termo \303\251 o que se procura: uma integral igual a p1, por\303\251m com uma pot\303\252ncia menor. Lamentavelmente, a integral original, p1, aparece novamente no \303\272ltimo termo. Contudo, este problema ainda pode ser contornado, resolvendo-se a equa\303\247\303\243o resultante para p1. Como o Maple considera que p1 j\303\241 possui um valor, provavelmente se recusar\303\241 a resolver a equa\303\247\303\243o. Neste caso, \303\251 preciso substituir p1 na equa\303\247\303\243o por uma express\303\243o n\303\243o atribu\303\255da, e ent\303\243o resolver para esta express\303\243o.

eqn1 := p1 = p6;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcW4xR0YoLy1JJEludEdGJzYkKS1JJHNpbkdGJTYjSSJ4R0YoSSJuR0YoRjYsKC1GMDYkKUYzLCZGNyIiIiEiI0Y9RjZGPSooLUkkY29zR0YlRjVGPSlGMywmRjdGPSEiIkY9Rj1GQ0ZERkQqJkZDRkRGL0Y9RkQ3I0Yu

eqn2 := subs(p1=a, eqn1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcW4yR0YoL0kiYUdGKCwoLUkkSW50R0YnNiQpLUkkc2luR0YlNiNJInhHRigsJkkibkdGKCIiIiEiI0Y7RjhGOyooLUkkY29zR0YlRjdGOylGNSwmRjpGOyEiIkY7RjtGQUZCRkIqJkZBRkJGL0Y7RkI3I0Yu

p7 := solve(eqn2, a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwN0dGKCwkKiYsKComLUkkY29zR0YlNiNJInhHRigiIiIpLUkkc2luR0YlRjQsJkkibkdGKEY2ISIiRjZGNkY2KiYtSSRJbnRHRic2JClGOCwmRjtGNiEiI0Y2RjVGNkY7RjZGPEY+RjZGNkY7RjxGPDcjRi4=

p1 = p7 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQpLUkkc2luR0YlNiNJInhHRihJIm5HRihGNCwmKiYsKComLUkkY29zR0YlRjMiIiIpRjEsJkY1RjwhIiJGPEY8RjwqJi1GLjYkKUYxLCZGNUY8ISIjRjxGNEY8RjVGPEY/RkFGPEY8RjVGP0Y/SSJDR0YoRjxGKw==

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

Utilizar a f\303\263rmula de redu\303\247\303\243o para 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 no c\303\241lculo das anti-derivadas de 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GIzYjLUklbXN1cEdGJDYlLUYjNiUtRiw2JVEkc2luRicvJSdpdGFsaWNHRkBGOC1GMjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y4Rj5GQUZDRkVGR0ZJRktGTUZQLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEieEYnL0ZnblEldHJ1ZUYnL0Y5USdpdGFsaWNGJ0Y4LUkjbW5HRiQ2JFEiNEYnRjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGY3AvJSpsaW5lYnJlYWtHUSVhdXRvRictRjI2L1EwJkRpZmZlcmVudGlhbEQ7RidGNUY4RjtGPkZBRkNGRUZHRklGS0ZNRlBGYG9GKw== 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 .

Pode-se resolver manualmente, ou pode-se ainda escrever um procedimento no Maple que o fa\303\247a. Por exemplo, pode-se escrever um procedimento que receba como entrada o expoente n, e que gere como sa\303\255da o lado direito da f\303\263rmula de redu\303\247\303\243o. Melhor ainda: com um loop for, ou um programa recursivo, pode-se escrever um procedimento que recebe n como entrada e resolve completamente a integral.

Solu\303\247\303\265es. Resolveremos as duas primeiras derivadas atrav\303\251s de um procedimento cuja entrada \303\251 n, e cuja sa\303\255da \303\251 o lado direito da f\303\263rmula. Note o uso do comando op (explicado na se\303\247\303\243o opcional acima) para a separa\303\247\303\243o dos termos nas respostas intermedi\303\241rias.

sinred := proc(n) (-1/n)*sin(x)^(n-1) * cos(x)

+ (n-1)/n * Int( sin(x)^(n-2), x);

end:

A solu\303\247\303\243o para 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 \303\251 dada por:

p1 := Int( sin(x)^5 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQtSSRzaW5HRiU2I0kieEdGKCIiJkY1NyNGLg==

p2 := sinred(5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiJS1JJGNvc0dGJUYyIiIiIyEiIiIiJi1JJEludEdGJzYkKiRGMCIiJEYzI0Y0Rjo3I0Yu

p3 := op(1, p2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwkKiYtSSRzaW5HRiU2I0kieEdGKCIiJS1JJGNvc0dGJUYyIiIiIyEiIiIiJjcjRi4=

p4 := (4/5)*sinred(3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiIy1JJGNvc0dGJUYyIiIiIyEiJSIjOi1JJEludEdGJzYkRjBGMyMiIilGOjcjRi4=

p5 := p3 + value(p4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNUdGKCwoKiYtSSRzaW5HRiU2I0kieEdGKCIiJS1JJGNvc0dGJUYyIiIiIyEiIiIiJiomRjAiIiNGNUY3IyEiJSIjOkY1IyEiKUY/NyNGLg==

p1 = p5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJC1JJHNpbkdGJTYjSSJ4R0YoIiImRjQsKiomRjEiIiUtSSRjb3NHRiVGMyIiIiMhIiJGNSomRjEiIiNGOUY7IyEiJSIjOkY5IyEiKUZCSSJDR0YoRjtGKw==

Tem-se agora 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 , organizado de uma forma ligeiramente diferente:

q1 := Int( sin(x)^4 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiQtSSRzaW5HRiU2I0kieEdGKCIiJUY1NyNGLg==

q2 := sinred(4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiJC1JJGNvc0dGJUYyIiIiIyEiIiIiJS1JJEludEdGJzYkKiRGMCIiI0YzI0Y0Rjo3I0Yu

q3 := (3/4)*sinred(2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiIi1JJGNvc0dGJUYyRjQjISIkIiIpLUkkSW50R0YnNiRGNEYzIyIiJEY5NyNGLg==

q4 := value(q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiIi1JJGNvc0dGJUYyRjQjISIkIiIpRjMjIiIkRjk3I0Yu

q5 := op(1, q2) + q4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKCwoKiYtSSRzaW5HRiU2I0kieEdGKCIiJC1JJGNvc0dGJUYyIiIiIyEiIiIiJSomRjBGN0Y1RjcjISIkIiIpRjMjRjRGPjcjRi4=

q1 = q5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJC1JJHNpbkdGJTYjSSJ4R0YoIiIlRjQsKiomRjEiIiQtSSRjb3NHRiVGMyIiIiMhIiJGNSomRjFGO0Y5RjsjISIkIiIpRjQjRjhGQUkiQ0dGKEY7Ris=

A anti-derivada 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 pode ser resolvida da mesma forma, embora seja um processo um pouco mais longo. Descrevemos a seguir um procedimento recursivo (um procedimento que chama a si mesmo repetidamente) para calcular a f\303\263rmula de redu\303\247\303\243o quantas vezes sejam necess\303\241rias. O procedimento abaixo \303\251 bastante b\303\241sico: uma vers\303\243o mais sofisticada poderia ainda verificar os valores de n, de forma a recusar valores do tipo 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 ou 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. O programa verifica se n vale, no m\303\255nimo, 2; neste caso, aplica-se a f\303\263rmula de redu\303\247\303\243o, que envolve uma chamada recursiva com um valor menor para n; caso contr\303\241rio, n \303\251 presumivelmente 0 ou 1, e a integral pode ser facilmente resolvida.

sinpowerint := proc(n)

if(n >= 2) then (-1/n)*sin(x)^(n-1) * cos(x)

+ (n-1)/n * sinpowerint(n-2);

else value(Int(sin(x)^n, x));

fi;

end:

Pode-se facilmente verificar o procedimento atrav\303\251s da resolu\303\247\303\243o de dois casos j\303\241 estudados:

sinpowerint(5);

LCgqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YpIiIlLUkkY29zR0YmRioiIiIjISIiIiImKiZGJCIiI0YtRi8jISIlIiM6Ri0jISIpRjc=

sinpowerint(4);

LCgqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YpIiIkLUkkY29zR0YmRioiIiIjISIiIiIlKiZGJEYvRi1GLyMhIiQiIilGKyNGLEY2

As respostas est\303\243o em acordo com o que foi obtido anteriormente. Pode-se ent\303\243o proceder com os c\303\241lculos:

Int( sin(x)^19, x) = sinpowerint(19) + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJC1JJHNpbkdGJTYjSSJ4R0YoIiM+RjQsOComRjEiIz0tSSRjb3NHRiVGMyIiIiMhIiJGNSomRjEiIztGOUY7IyEjPSIkQiQqJkYxIiM5RjlGOyMhIycqIiU6OyomRjEiIzdGOUY7IyElVzgiJiYqNCMqJkYxIiM1RjlGOyMhJkdoIiInWDRCKiZGMSIiKUY5RjsjISUlZSQiJio9WSomRjEiIidGOUY7IyElJzQlRlYqJkYxIiIlRjlGOyMhJndYI0ZRKiZGMSIiI0Y5RjsjISZvRiRGUUY5IyEmT2InRlFJIkNHRihGO0Yr

Parab\303\251ns! Voc\303\252 acaba de re-escrever uma pequena parte do comando int do Maple.

O pr\303\263ximo caso especial \303\251 a integral 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 . Este tipo de integral pode ser resolvida pela f\303\263rmula de redu\303\247\303\243o

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 + 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 .

Finalmente, para o caso geral, h\303\241 duas f\303\263rmulas:

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 = 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 + 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 ,

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 = 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 + 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 .

O uso repetido de uma ou de outra reduz o problema a um dos casos especiais j\303\241 estudados.

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

Calcular as integrais 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, 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 e 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.

Solu\303\247\303\265es. Para as duas primeiras, reduziremos a pot\303\252ncia do coseno. No primeiro exemplo, n\303\243o faz diferen\303\247a que dire\303\247\303\243o seguir, apesar de o coseno possuir a menor pot\303\252ncia; no segundo caso, \303\251 mais r\303\241pido reduzir a pot\303\252ncia \303\255mpar que a pot\303\252ncia par. A fim de simplificar o trabalho, escreveremos um procedimento que reduz a pot\303\252ncia do coseno. n \303\251 a pot\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Yr; m \303\251 a pot\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Yr .

lowercos := proc(n,m) sin(x)^(n+1)*cos(x)^(m-1)/(n+m)

+ (m-1)/(n+m)*Int(sin(x)^n * cos(x)^(m-2) , x);

end:

p1 := Int(sin(x)^7 * cos(x)^5 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiYtSSRzaW5HRiU2I0kieEdGKCIiKC1JJGNvc0dGJUY0IiImRjU3I0Yu

p2 := lowercos(7,5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiKS1JJGNvc0dGJUYyIiIlIyIiIiIjNy1JJEludEdGJzYkKiZGMCIiKEY1IiIkRjMjRjlGQDcjRi4=

p3 := (1/3)*lowercos(7,3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiKS1JJGNvc0dGJUYyIiIjIyIiIiIjSS1JJEludEdGJzYkKiZGMCIiKEY1RjlGMyNGOSIjOjcjRi4=

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiKS1JJGNvc0dGJUYyIiIjIyIiIiIjSSokRjBGNCNGOSIkPyI3I0Yu

p1 = op(1,p2) + p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHNpbkdGJTYjSSJ4R0YoIiIoLUkkY29zR0YlRjMiIiZGNCwqKiZGMSIiKUY2IiIlIyIiIiIjNyomRjFGO0Y2IiIjI0Y+IiNJKiRGMUY7I0Y+IiQ/IkkiQ0dGKEY+Ris=

(\303\211 interessante fazer a verifica\303\247\303\243o do resultado por diferencia\303\247\303\243o!)

Consideramos agora o segundo exemplo. Note que, com a escolha da pot\303\252ncia \303\255mpar, obt\303\251m-se uma anti-derivada bem mais simples, de forma que a outra pot\303\252ncia n\303\243o precisar\303\241 ser reduzida.

q1 := Int( sin(x)^4 * cos(x)^7 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiYtSSRzaW5HRiU2I0kieEdGKCIiJS1JJGNvc0dGJUY0IiIoRjU3I0Yu

q2 := lowercos(4,7);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiJi1JJGNvc0dGJUYyIiInIyIiIiIjNi1JJEludEdGJzYkKiZGMCIiJUY1RjRGMyNGN0Y6NyNGLg==

q3 := (6/11)*lowercos(4,5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiJi1JJGNvc0dGJUYyIiIlIyIiIyIjTC1JJEludEdGJzYkKiZGMEY3RjUiIiRGMyMiIilGOjcjRi4=

q4 := (8/33)*lowercos(4,3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiJi1JJGNvc0dGJUYyIiIjIyIiKSIkSiMtSSRJbnRHRic2JComRjAiIiVGNSIiIkYzIyIjO0Y6NyNGLg==

A \303\272ltima anti-derivada pode ser resolvida pela substitui\303\247\303\243o de 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; considerando as respostas anteriores, o Maple pode resolver esta anti-derivada em um \303\272nico passo.

q5 := value(q4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNUdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiJi1JJGNvc0dGJUYyIiIjIyIiKSIkSiMqJEYwRjQjIiM7IiViNjcjRi4=

Em seguida, agrupamos todos os peda\303\247os:

q1 = op(1, q2) + op(1, q3) + q5 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHNpbkdGJTYjSSJ4R0YoIiIlLUkkY29zR0YlRjMiIihGNCwsKiZGMSIiJkY2IiInIyIiIiIjNiomRjFGO0Y2RjUjIiIjIiNMKiZGMUY7RjZGQiMiIikiJEojKiRGMUY7IyIjOyIlYjZJIkNHRihGPkYr

No \303\272ltimo exemplo, como ambas as pot\303\252ncias s\303\243o pares, n\303\243o faz diferen\303\247a que dire\303\247\303\243o tomar. N\303\243o h\303\241 grande dificuldade em reduzir a menor pot\303\252ncia. H\303\241 uma outra vantagem neste caso: j\303\241 existe um procedimento que reduz a pot\303\252ncia do coseno, assim como um procedimento que resolve a integral resultante, depois que todas as pot\303\252ncias do coseno tiverem sido eliminadas.

r1 := Int( sin(x)^6 * cos(x)^4, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMUdGKC1JJEludEdGJzYkKiYtSSRzaW5HRiU2I0kieEdGKCIiJy1JJGNvc0dGJUY0IiIlRjU3I0Yu

r2 := lowercos(6,4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMkdGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiKC1JJGNvc0dGJUYyIiIkIyIiIiIjNS1JJEludEdGJzYkKiZGMCIiJ0Y1IiIjRjMjRjdGOjcjRi4=

r3 := (3/10)*lowercos(6,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyM0dGKCwmKiYtSSRzaW5HRiU2I0kieEdGKCIiKC1JJGNvc0dGJUYyIiIiIyIiJCIjISktSSRJbnRHRic2JCokRjAiIidGM0Y4NyNGLg==

r4 := (3/80)*sinpowerint(6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyNEdGKCwqKiYtSSRzaW5HRiU2I0kieEdGKCIiJi1JJGNvc0dGJUYyIiIiIyEiIiIkZyIqJkYwIiIkRjVGNyNGOSIkRyIqJkYwRjdGNUY3IyEiJCIkYyNGMyNGPEZCNyNGLg==

r1 = op(1,r2) + op(1,r3) + r4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHNpbkdGJTYjSSJ4R0YoIiInLUkkY29zR0YlRjMiIiVGNCwwKiZGMSIiKEY2IiIkIyIiIiIjNSomRjFGO0Y2Rj4jRjwiIyEpKiZGMSIiJkY2Rj4jISIiIiRnIiomRjFGPEY2Rj4jRkYiJEciKiZGMUY+RjZGPiMhIiQiJGMjRjQjRjxGTkkiQ0dGKEY+Ris=

<Text-field style="Heading 1" layout="Heading 1">Integrais da forma <Equation executable="false" style="2D Comment" input-equation="" display="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">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</Equation></Text-field>

Integrais deste tipo podem ser tratadas de forma muito similar \303\240quelas da se\303\247\303\243o anterior. Os casos especiais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw== ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw== podem ser tratados pelas f\303\263rmulas de redu\303\247\303\243o

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 = 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 - 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 = 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 + 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

respectivamente. O caso geral pode ser resolvido por uma das f\303\263rmulas

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 = 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 - 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHRjMvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJw== 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 - 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 .

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

Encontrar as anti-derivadas 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, 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 , 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 e 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 .

Solu\303\247\303\265es. Resolveremos as duas primeiras atrav\303\251s de procedimentos para redu\303\247\303\243o das pot\303\252ncias da tangente ou secante.

tanred := proc(n) 1/(n-1) * tan(x)^(n-1) - Int(tan(x)^(n-2), x) end:

secred := proc(m) 1/(m-1) * tan(x)*sec(x)^(m-2)

+ (m-2)/(m-1) * Int(sec(x)^(m-2) , x) end:

p1 := Int( tan(x)^4, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJEludEdGJzYkKiQtSSR0YW5HRiU2I0kieEdGKCIiJUY1NyNGLg==

p2 := tanred(4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMkdGKCwmKiQtSSR0YW5HRiU2I0kieEdGKCIiJCMiIiJGNC1JJEludEdGJzYkKiRGMCIiI0YzISIiNyNGLg==

p3 := -tanred(2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwM0dGKCwmLUkkdGFuR0YlNiNJInhHRighIiItSSRJbnRHRic2JCIiIkYyRjc3I0Yu

p4 := value(p3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwNEdGKCwmLUkkdGFuR0YlNiNJInhHRighIiJGMiIiIjcjRi4=

p1 = op(1, p2) + p4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJC1JJHRhbkdGJTYjSSJ4R0YoIiIlRjQsKiokRjEiIiQjIiIiRjhGMSEiIkY0RjpJIkNHRihGOkYr

q1 := Int( sec(x)^5, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC1JJEludEdGJzYkKiQtSSRzZWNHRiU2I0kieEdGKCIiJkY1NyNGLg==

q2 := secred(5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKCwmKiYtSSR0YW5HRiU2I0kieEdGKCIiIi1JJHNlY0dGJUYyIiIkI0Y0IiIlLUkkSW50R0YnNiQqJEY1RjdGMyNGN0Y5NyNGLg==

q3 := (3/4)*secred(3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKCwmKiYtSSR0YW5HRiU2I0kieEdGKCIiIi1JJHNlY0dGJUYyRjQjIiIkIiIpLUkkSW50R0YnNiRGNUYzRjc3I0Yu

Neste caso, a f\303\263rmula de redu\303\247\303\243o j\303\241 n\303\243o tem utilidade. Basta resolver normalmente a anti-derivada, que j\303\241 \303\251 bastante conhecida.

q4 := value(q3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKCwmKiYtSSR0YW5HRiU2I0kieEdGKCIiIi1JJHNlY0dGJUYyRjQjIiIkIiIpLUkjbG5HRiU2IywmRjVGNEYwRjRGNzcjRi4=

q1 = op(1, q2) + q4 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJC1JJHNlY0dGJTYjSSJ4R0YoIiImRjQsKiomLUkkdGFuR0YlRjMiIiJGMSIiJCNGOiIiJSomRjhGOkYxRjojRjsiIiktSSNsbkdGJTYjLCZGMUY6RjhGOkY/SSJDR0YoRjpGKw==

Nos dois \303\272ltimos exemplos, note que, se iniciarmos pela redu\303\247\303\243o da pot\303\252ncia da tangente a cada vez, terminaremos (ap\303\263s duas aplica\303\247\303\265es da f\303\263rmula de redu\303\247\303\243o apropriada) com uma integral que envolve apenas secante. A maneira mais simples de se resolver o problema \303\251 criando dois programas recursivos: um que reduz a pot\303\252ncia da tangente, e outro que resolve a anti-derivada de uma pot\303\252ncia da secante. Eis aqui (n \303\251 a pot\303\252ncia da tangente; m \303\251 a pot\303\252ncia da secante).

tanred2 := proc(n,m)

if(n >= 2) then (1/m)*tan(x)^(n-1)*sec(x)^m

- (n-1)/m * tanred2(n-2,m+2);

else Int(tan(x)^n * sec(x)^m, x);

fi;

end:

secpowerint := proc(m)

if(m >= 2) then 1/(m-1)*tan(x)*sec(x)^(m-2)

+ (m-2)/(m-1) * secpowerint(m-2);

else value(Int(sec(x)^m, x));

fi;

end:

Pode-se agora resolver ambas as anti-derivadas.

r1 := Int(tan(x)^4 * sec(x)^4 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMUdGKC1JJEludEdGJzYkKiYtSSR0YW5HRiU2I0kieEdGKCIiJS1JJHNlY0dGJUY0RjZGNTcjRi4=

r2 := tanred2(4,4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyMkdGKCwoKiYtSSR0YW5HRiU2I0kieEdGKCIiJC1JJHNlY0dGJUYyIiIlIyIiIkY3KiZGMEY5RjUiIicjISIiIiIpLUkkSW50R0YnNiQqJEY1Rj5GMyNGOUY+NyNGLg==

r3 := (3/8)*secpowerint(8);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNyM0dGKCwqKiYtSSR0YW5HRiU2I0kieEdGKCIiIi1JJHNlY0dGJUYyIiInIyIiJCIjYyomRjBGNEY1IiIlIyIiKiIkUyIqJkYwRjRGNSIiIyNGOSIjTkYwI0Y3RkM3I0Yu

r1 = op(1,r2) + op(2,r2) + r3 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHRhbkdGJTYjSSJ4R0YoIiIlLUkkc2VjR0YlRjNGNUY0LC4qJkYxIiIkRjZGNSMiIiJGNSomRjFGPEY2IiInIyEiIiIjOSomRjFGPEY2RjUjIiIqIiRTIiomRjFGPEY2IiIjI0Y6IiNORjEjRj5GSUkiQ0dGKEY8Ris=

s1 := Int( tan(x)^4 * sec(x)^5 , x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNzMUdGKC1JJEludEdGJzYkKiYtSSR0YW5HRiU2I0kieEdGKCIiJS1JJHNlY0dGJUY0IiImRjU3I0Yu

s2 := tanred2(4,5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNzMkdGKCwoKiYtSSR0YW5HRiU2I0kieEdGKCIiJC1JJHNlY0dGJUYyIiImIyIiIkY3KiZGMEY5RjUiIigjISIkIiNOLUkkSW50R0YnNiQqJEY1IiIqRjMjRjRGPjcjRi4=

s3 := (3/35)*secpowerint(9);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNzM0dGKCwsKiYtSSR0YW5HRiU2I0kieEdGKCIiIi1JJHNlY0dGJUYyIiIoIyIiJCIkIUcqJkYwRjRGNSIiJiNGNCIjISkqJkYwRjRGNUY5I0Y0IiNrKiZGMEY0RjVGNCNGOSIkRyItSSNsbkdGJTYjLCZGNUY0RjBGNEZDNyNGLg==

s1 = op(1,s2) + op(2,s2) + s3 + C;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkSW50R0YnNiQqJi1JJHRhbkdGJTYjSSJ4R0YoIiIlLUkkc2VjR0YlRjMiIiZGNCwwKiZGMSIiJEY2RjgjIiIiRjgqJkYxRj1GNiIiKCMhIiQiI1MqJkYxRj1GNkY4I0Y9IiMhKSomRjFGPUY2RjsjRj0iI2sqJkYxRj1GNkY9I0Y7IiRHIi1JI2xuR0YlNiMsJkY2Rj1GMUY9RkpJIkNHRihGPUYr