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Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
Cálculo 2: Aula 2 - Antiderivadas
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Cálculo 2: Aula 2 - Antiderivadas

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  • 1. Aula 2 - Anti-diferencia¸˜o caWillian Vieira de PaulaRegra daCadeia paraAntiderivadas 1 10 Exemplo: Diferenciar 10 1 + x2 .
  • 2. Aula 2 - Anti-diferencia¸˜o caWillian Vieira de PaulaRegra daCadeia paraAntiderivadas 1 10 Exemplo: Diferenciar 10 1 + x 2 . Fa¸a o processo oposto, ou seja, antidiferencie 1 + x 2 )9 (2x) . c
  • 3. Aula 2 - Anti-diferencia¸˜o caWillian Vieira de PaulaRegra daCadeia para TeoremaAntiderivadas Seja g uma fun¸˜o diferenci´vel e seja o intervalo I a imagem ca a de g . Suponha que f seja uma fun¸˜o definida em I e que F ca seja uma primitiva de f em I . Ent˜o a f g (x)) g (x)dx = F (g (x)) + C
  • 4. Aula 2 - Anti-diferencia¸˜o caWillian Vieira de PaulaRegra daCadeia para Corol´rio aAntiderivadas Se g for uma fun¸˜o diferenci´vel e se n for um n´mero ca a u racional, [g (x)]n+1 [g (x)]n g (x)dx = + C , n = −1. n+1
  • 5. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia paraAntiderivadas
  • 6. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia para Ache x 2 (3 + 2x 3 )8 dxAntiderivadas
  • 7. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia para Ache x 2 (3 + 2x 3 )8 dxAntiderivadas Calcule xcos(x 2 )dx
  • 8. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia para Ache x 2 (3 + 2x 3 )8 dxAntiderivadas Calcule xcos(x 2 )dx 2x − 9 Calcule √ dx x2− 9x + 1
  • 9. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia para Ache x 2 (3 + 2x 3 )8 dxAntiderivadas Calcule xcos(x 2 )dx 2x − 9 Calcule √ dx x2 − 9x + 1 √ Calcule x 2 1 − xdx
  • 10. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia para Ache x 2 (3 + 2x 3 )8 dxAntiderivadas Calcule xcos(x 2 )dx 2x − 9 Calcule √ dx x2 − 9x + 1 √ Calcule x 2 1 − xdx Calcule cos 2 (x)dx
  • 11. Aula 2 - Anti- Exemplosdiferencia¸˜o ca √Willian Vieira de Paula Calcule 3x + 4dxRegra daCadeia para Ache x 2 (3 + 2x 3 )8 dxAntiderivadas Calcule xcos(x 2 )dx 2x − 9 Calcule √ dx x2 − 9x + 1 √ Calcule x 2 1 − xdx Calcule cos 2 (x)dx 1 Calcule dx cos 2 (2x)

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