Math118_S10_W3

Math118_S10_W3 - Math118(M Kohandel-Outline(week 3 3.1...

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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (week 3) 3.1 Partial fractions (8.6) 3.2 Numerical integration (8.7) Rectangular rule Trapezoidal rule Simpson's rule
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 3.1 Partial fractions In this lecture, we will consider the integration of rational functions, i.e. functions that are the ratios of polynomials; for example, dx x x x 5 6 7 3 2 , C x x dx x x | 1 | ln | 5 | ln 2 ] 1 1 5 2 [ We note that 5 6 7 3 ) 1 )( 5 ( ) 5 ( ) 1 ( 2 1 1 5 2 2 x x x x x x x x x . This example illustrate a procedure for integrating certain rational functions. In this case, we need to reverse the process, i.e. starting from a rational function, we break it down into simpler component fractions called "partial fractions". Consider ) ( / ) ( ) ( x q x p x f , where ) ( x p and ) ( x q are polynomials in x . We assume that the degree of ) ( x p is less than degree of ) ( x q . [Note that if the degree of ) ( x p is greater or equal to degree of ) ( x q , we can apply long division; we will do some examples later]. Case 1: If the denominator ) ( x q contains a product of n distinct linear factors, ) ( ) )( ( 2 2 1 1 n n b x a b x a b x a where i a and i b are real numbers, then unique real constants i c can be found such that n n n b x a c b x a c b x a c x q x p x f 2 2 2 1 1 1 ) ( ) ( ) ( Example 3.1: Evaluate dx x x x dx x x x ) 3 )( 1 ( 1 2 3 2 1 2 2 (check the degrees). ). 5 ( ) 4 ( , 3 4 ) 1 ( in 3 and 1 Set : Shortcut | 3 | ln 4 5 | 1 | ln 4 3 3 4 5 1 4 3 ) 3 )( 1 ( 1 2 ] | | ln 1 : Note [ 3 ) 4 / 5 ( 1 ) 4 / 3 ( ) 3 )( 1 ( 1 2 4 / 5 , 4 / 3 1 3 , 2 1 2 ) 3 ( ) ( ) 1 ( 1 2 ) 1 ( ) 3 ( ) 3 )( 1 ( ) 1 ( ) 3 ( 3 1 ) 3 )( 1 ( 1 2 B A x x C x x x dx x dx dx x x x C b ax a b ax dx x x x x x B A B A B A x B A x B A x x B x A x x x B x A x B x A x x x
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 3 Case 2: If the denominator ) ( x q contains a repeated linear factor n b ax ) ( , then n n b ax c b ax c b ax c x q x p x f ) ( ) ( ) ( ) ( ) ( 2 2 1 Example 3.2: Evaluate dx x x x 3 2 ) 1 ( 4 2 . )] 1 , 0 ( 1 ) ( 1 ) ( ) ( : [Note ) 1 ( 2 3 | 1 | ln ] ) 1 ( 3 1 1 [ ) 1 ( 4 2 0 , 1 ) 3 ( ) 2 ( 4 2 3 1 ) 1 ( ) 1 ( 4 2 ) 1 ( ) 1 ( 1 ) 1 ( 4 2 1 1 1 2 3 3 2 2 2 2 2 3 2 3 2 n C n b ax a dx b ax b ax dx C x x dx x x dx x x x B A B A x B A Ax x x C x C x B x A x x x C x B x A x x x n n n When the denominator contains distinct as well as repeated linear factors, we combine two cases.
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This note was uploaded on 02/06/2011 for the course MATH 118 taught by Professor Zhou during the Spring '08 term at Waterloo.

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Math118_S10_W3 - Math118(M Kohandel-Outline(week 3 3.1...

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