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Section 11.5 Partial Fraction Decomposition
Section 11.5Partial Fraction Decomposition
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2
7 1Write the partial fraction decomposition of 6
xx x
( 3)( 2)x x 2
7 16 3 2
x A Bx x x x
2
( 3)( 2) ( 3)( 2) ( 37 1 )(6
23 2
)x x x x x xx A Bx x x x
7 1 ( 2) ( 3)x A x B x 2 3A B x A B
Equate coefficients of like powers of x to get: 71 2 3
A BA B
4 3A B
2
7 1 4 36 3 2
xx x x x
2
2
2Write the partial fraction decomposition of 1 2x
x x
2
2 2
21 21 2 2
x A B Cx xx x x
2Clear of fractions by multiplying all terms by 1 2x x
22 2 2 1 2 1x A x B x x C x
2 2 22 4 4 2x Ax Ax A Bx Bx B Cx C
2 22 4 4 2x A B x A B C x A B C
10 42 4 2
A BA B CA B C
1 2, , 23 3
A B C
2
1 223 3
1 2 2x x x
2
23
4Write the partial fraction decomposition of 1
xx x
2
2 22 33
411 1
x A B C D Ex x x xx x x
2 2 22 2 3 34 1 1 1 1x Ax x Bx x C x Dx x Ex
2Let 0: 4 0 1x C 4C 32Let 1: 1 4 1x E 3E
2 2 22 3 2 34 4 1 3 1 1 1x x x Ax x Bx x Dx x
23 8 ( 1) 1 1 1x x x x x Ax x B x Dx
23 8 1 1x Ax x B x Dx Let 0 :8 0 1x B 8B
2Let 1: 3(1) 8 1x D 11D
23 8 1 8 1 11x Ax x x x
2Let 2 :3 2 8 2 2 1 8 2 1 11 2x A 11A
2
2 22 33
4 11 8 4 11 311 1
xx x x xx x x
2
1Write the partial fraction decomposition of 1 4x x
22
11 41 4A Bx Cx xx x
21 4 1A x Bx C x
2 21 4Ax A Bx Bx Cx C
21 4A B x B C x A C
001 4
A BB CA C
15
15
15
A
B
C
2
1 1 15 5 5
1 4
x
x x
2
22
1Write the partial fraction decomposition of 4
x
x
2
2 222 2
144 4
x Ax B Cx Dxx x
2 21 4x Ax B x Cx D
2 3 21 4 4x Ax Bx Ax B Cx D
2 3 21 4 4x Ax Bx A C x B D
010 41 4
ABA CB D
010
3
ABCD
22 2
1 34 4x x
