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# Calculus II - 4

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Stewart Calculus Section 7.4

Stewart Calculus Section 7.4

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• ### Transcript

• 1. 7.4 Integration of Rational FunctionsRational function: ratio of polynomials.Ex: + + +
• 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. 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. 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. Second step: factor the dominator of theproper rational function to be a product oflinear factors and/or irreducible quadraticfactors. + + is irreducible if < .Ex: + = ( + ) + = ( )( + ) + + =( ) ( + )
• 6. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +
• 7. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + )
• 8. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + ) +so assume = + + + +
• 9. Third step: decompose the proper rationalfunction to be a sum of partial fractions bymethod of undetermined coefficients. +Ex: +We factor + = ( )( + ) +so assume = + + + + ( + + ) +( + ) = +
• 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. 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. +Ex: find +
• 13. +Ex: find +We already knew that + = + + ( ) ( + ) C AS E I
• 14. +Ex: find +We already knew that + = + + ( ) ( + ) + C AS E Iso + = | |+ | | | + |+
• 15. Ex: find +
• 16. Ex: find +We factor that + =( ) ( + ) CAS E II
• 17. Ex: find +We factor that + =( ) ( + ) CAS E IIso we assume = + + + ( ) +
• 18. Ex: find +We factor that + =( ) ( + ) CAS E IIso we assume = + + + ( ) + and find = , = , =
• 19. Ex: find +We factor that + =( ) ( + ) CAS E IIso we assume = + + + ( ) + and find = , = , =thus + = | | | + |+