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Cálculo II - Aula 3: Integração por partes

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The document presents the derivation of the integration by parts formula. It shows that the formula can be derived by taking the derivative of the product of two functions f(x) and g(x), and then integrating both sides. This results in the integration by parts formula: the integral of f(x)g(x)dx is equal to f(x)g(x) - the integral of f'(x)g(x)dx. An example is then given of using integration by parts to calculate the integral of xcosxdx.

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Observe que x 2 dx = xdx · xdx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) d “Integrando” dx [f (x)g (x)] dx = [f (x)g (x) + f (x)g (x)] dx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) d “Integrando” dx [f (x)g (x)] dx = [f (x)g (x) + f (x)g (x)] dx d f (x)g (x)dx = dx [f (x)g (x)] dx − f (x)g (x)dx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) d “Integrando” dx [f (x)g (x)] dx = [f (x)g (x) + f (x)g (x)] dx d f (x)g (x)dx = dx [f (x)g (x)] dx − f (x)g (x)dx Portanto f (x)g (x)dx = f (x)g (x) − f (x)g (x)dx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
Exemplo: Calcule, usando integra¸˜o por partes: ca xcosxdx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
¸˜ INTEGRACAO POR PARTES udv = uv − vdu Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca

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Cálculo II - Aula 3: Integração por partes

  • 1. Observe que x 2 dx = xdx · xdx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
  • 2. d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
  • 3. d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) d “Integrando” dx [f (x)g (x)] dx = [f (x)g (x) + f (x)g (x)] dx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
  • 4. d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) d “Integrando” dx [f (x)g (x)] dx = [f (x)g (x) + f (x)g (x)] dx d f (x)g (x)dx = dx [f (x)g (x)] dx − f (x)g (x)dx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
  • 5. d dx [f (x)g (x)] = f (x)g (x) + f (x)g (x) d “Integrando” dx [f (x)g (x)] dx = [f (x)g (x) + f (x)g (x)] dx d f (x)g (x)dx = dx [f (x)g (x)] dx − f (x)g (x)dx Portanto f (x)g (x)dx = f (x)g (x) − f (x)g (x)dx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
  • 6. Exemplo: Calcule, usando integra¸˜o por partes: ca xcosxdx Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca
  • 7. ¸˜ INTEGRACAO POR PARTES udv = uv − vdu Willian Vieira de Paula Aula 3 - Integra¸˜o por partes ca