Explanation Given: I = ∫ P ( x ) Q ( x ) d x If P ( x ) Q ( x ) is a proper rational function where Q ( x ) factors as product of distinct linear factors ( x − a i ) . Then the integral I = ∫ P ( x ) Q ( x ) d x (a) is the sum of logarithmic terms A i ln ( x − a i ) for some constants A i . (b) May contain a term involving the arctangent. Solution: I = ∫ P ( x ) Q ( x ) d x Where P ( x ) Q ( x ) is a proper rational function where Q ( x ) factors as product of distinct linear factors ( x − a i ) . Since, P ( x ) Q ( x ) is a proper rational function which infers that given rational functions do not have common prime factors. As Q ( x ) factors as product of distinct linear factors ( x − a i ) . Q ( x ) = ( x − a 1 ) ( x − a 2 ) ( x − a 3 ) ... ( x − a n ) ⇒ Q ( x ) = ( x − a 1 ) ( x − a 2 ) ( x − a 3 ) ... ( x − a i ) ... ( x − a n ) ⇒ Q ( x ) = ∏ i = 1 n ( x − a i ) ⇒ P ( x ) Q ( x ) = P ( x ) ( x − a 1 ) ( x − a 2 ) ( x − a 3 ) ... ( x − a i ) ... ( x − a n ) ⇒ P ( x ) Q ( x ) = C 1 ( x − a 1 ) + C 2 ( x − a 2 ) + C 3 ( x − a 3 ) + ... + C i ( x − a i ) + ... + C n ( x − a n ) ( 1 ) ⇒ C 1 = lim x → a 1 ( x − a 1 ) P ( x ) ( x − a 1 ) ( x − a 2 ) ( x − a 3 ) ... ( x − a i ) ... ( x − a n ) ⇒ C 1 = lim x → a 1 P ( x ) ( x − a 2 ) ( x − a 3 ) ... ( x − a i ) ... ( x − a n ) ⇒ C 1 = P ( a 1 ) ( a 1 − a 2 ) ( a 1 − a 3 ) ... ( a 1 − a i ) ... ( a 1 − a n ) = P ( a 1 ) M 1 Similarly, C 2 = P ( a 2 ) ( a 2 − a 1 ) ( a 2 − a 3 ) ... ( a 2 − a i ) ... ( a 2 − a n ) = P ( a 2 ) M 2 C 3 = P ( a 3 ) ( a 3 − a 1 ) ( a 3 − a 3 ) ... ( a 3 − a i ) ... ( a 3 − a n ) = P ( a 3 ) M 3 C i = P ( a i ) ( a i − a 1 ) ( a i − a 3 ) ... ( a i − a i − 1 ) ... ( a i − a n ) = P ( a i ) M i C n = P ( a n ) ( a n − a 1 ) ( a n − a 2 ) ( a n − a 3 ) ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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