aula 31 calculo iialuno
TRANSCRIPT
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Professor: Carlos Alberto de Albuquerque
Blog: http://professorcarlosaa.blogspot.com.br/
Email:
CÁLCULO DIFERENCIAL E INTEGRAL II
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AULA
31
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
As substituições trigonométricas podem servir
para transformar integrais que envolvam
em integrais que podemos calcular
diretamente.
222222 , axexaxa
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
TRÊS SUBSTITUIÇÕES BÁSICAS
As substituições mais comuns são
Elas vem dos triângulos de referência da Figura
seguinte.
.sec, axesenaxtgax
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
A restrição de θ se deve ao foto que a função
deve ser invertível.
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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EXERCÍCIO 1
Calcule, usando uma substituição
trigonométrica, a integral
29 y
dy
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SOLUÇÃO
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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EXERCÍCIO 2
Calcule, usando uma substituição
trigonométrica, a integral
2
3
029 x
dx
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SOLUÇÃO
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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EXERCÍCIO 3
Calcule, usando uma substituição
trigonométrica, a integral
494 2x
dx
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SOLUÇÃO
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
Se a=b=r, concluímos que a área de um
círculo com raio r é πr2.
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
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Assim temos:
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SUBSTITUIÇÕES TRIGONOMÉTRICAS
Exemplo.
Calcule
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SOLUÇÃO
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EXERCÍCIO 4
Calcule
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SOLUÇÃO
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EXERCÍCIO 5
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SOLUÇÃO
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EXERCÍCIO 6
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SOLUÇÃO
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EXERCÍCIO 7
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SOLUÇÃO
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FIM
DA AULA
31