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Calculus II - 5

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Stewart Calculus Section 7.4

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Calculus II - 5

  1. 1. 7.4 Integration of Rational FunctionsFirst step: if the rational function isimproper, decompose it to a polynomial plusa proper rational function.Second step: factor the dominator to be aproduct of linear factors and/or irreduciblequadratic factors.Third step: decompose the proper rationalfunction to be a sum of partial fractions.
  2. 2. CASE I: the dominator is a product ofdistinct linear factors.CASE II: the dominator is a product oflinear factors, some of which are repeated.CASE III: the dominator contains distinctirreducible quadratic factors.Case IV: the dominator contains repeatedirreducible quadratic factors.
  3. 3. Ex: find +
  4. 4. Ex: find + I CA S E IIWe recognize + is irreducible.
  5. 5. Ex: find + I CA S E IIWe recognize + is irreducible.Complete the square: + =( ) +
  6. 6. Ex: find + I CA S E II We recognize + is irreducible. Complete the square: + =( ) += ( + ) − = + +
  7. 7. Ex: find + I CA S E II We recognize + is irreducible. Complete the square: + =( ) += ( + ) − = + + = + +
  8. 8. Ex: find + I CA S E II We recognize + is irreducible. Complete the square: + =( ) += ( + ) − = + + = + + = ( + ) +
  9. 9. Ex: find + I CA S E II We recognize + is irreducible. Complete the square: + =( ) += ( + ) − = + + = + + = ( + ) + = ( + ) +
  10. 10. +Ex: find +
  11. 11. +Ex: find + + +We assume = + CAS E III + +
  12. 12. +Ex: find + + +We assume = + CAS E III + +and solve = , = , = .
  13. 13. +Ex: find + + +We assume = + CAS E III + +and solve = , = , = . +so = + + + +
  14. 14. +Ex: find + + +We assume = + CAS E III + +and solve = , = , = . +so = + + + + = | |+ ( + ) +
  15. 15. + +Ex: find ( + )
  16. 16. + +Ex: find ( + ) CAS E IV + + + +We assume = + ( + ) + ( + )
  17. 17. + +Ex: find ( + ) CAS E IV + + + +We assume = + ( + ) + ( + )and solve = , = , = , = .
  18. 18. + +Ex: find ( + ) CAS E IV + + + +We assume = + ( + ) + ( + )and solve = , = , = , = . + + +so = ( + ) + ( + )
  19. 19. + +Ex: find ( + ) CAS E IV + + + +We assume = + ( + ) + ( + )and solve = , = , = , = . + + +so = ( + ) + ( + ) = ( + )+ + + ( + )
  20. 20. +Ex: find
  21. 21. +Ex: find √Let = +
  22. 22. +Ex: find √Let = + zation Rati onali +then =
  23. 23. +Ex: find √Let = + zation Rati onali +then = = +
  24. 24. +Ex: find √Let = + zation Rati onali +then = = + = + | | | + |+
  25. 25. +Ex: find √Let = + zation Rati onali +then = = + = + | | | + |+ + = + + + + +

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