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# Calculus II - 4

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Stewart Calculus Section 7.4

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### Calculus II - 4

1. 1. 7.4 Integration of Rational FunctionsRational function: ratio of polynomials.Ex: + + +
2. 2. 7.4 Integration of Rational FunctionsRational function: ratio of polynomials.Ex: + + +Fact: A rational function can always bedecomposed to summation of terms like +polynomials or ( + ) or ( + + ) whereA, B, etc. are constants.Ex: − + + = + + + − − − + − ( − ) + Partial Fractions
3. 3. First step: if the rational function isimproper (the degree of numerator is morethan or equal to the degree of dominator),use long division to decompose it to apolynomial plus a proper rational function. +Ex: = + + +
4. 4. First step: if the rational function isimproper (the degree of numerator is morethan or equal to the degree of dominator),use long division to decompose it to apolynomial plus a proper rational function. +Ex: = + + +As a result: + = + + + | |+
5. 5. Second step: factor the dominator of theproper rational function to be a product oflinear factors and/or irreducible quadraticfactors. + + is irreducible if < .Ex: + = ( + ) + = ( )( + ) + + =( ) ( + )
6. 6. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +
7. 7. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + )
8. 8. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + ) +so assume = + + + +
9. 9. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + ) +so assume = + + + + ( + + ) +( + ) = +
10. 10. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + ) +so assume = + + + + ( + + ) +( + ) = +thus = , = , =
11. 11. 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.
12. 12. +Ex: find +
13. 13. +Ex: find +We already knew that + = + + ( ) ( + ) C AS E I
14. 14. +Ex: find +We already knew that + = + + ( ) ( + ) + C AS E Iso + = | |+ | | | + |+
15. 15. Ex: find +
16. 16. Ex: find +We factor that + =( ) ( + ) CAS E II
17. 17. Ex: find +We factor that + =( ) ( + ) CAS E IIso we assume = + + + ( ) +
18. 18. Ex: find +We factor that + =( ) ( + ) CAS E IIso we assume = + + + ( ) + and find = , = , =
19. 19. Ex: find +We factor that + =( ) ( + ) CAS E IIso we assume = + + + ( ) + and find = , = , =thus + = | | | + |+