Thomas Calculus 13th [Solutions]

Section 8.5 Integration of Rational Functions by Partial Fractions 587
10. 1 A B 1 A( x 2) Bx; x 0 A 1 ; x 2 B 1 ;
2
x 2x
x x 2 2 2
dx 1 dx 1 dx 1 ln x ln x 2 C
2
x 2x
2 x 2 x 2 2
11. x 4 A B x 4 A( x 1) B( x 6); x 1 B 5 ; x 6 A 2 2 ;
2
x 5x
6 x 6 x 1 7 7 7
x 4 2 5 2 5
1 2 5
dx dx dx ln x 6 ln x 1 C ln ( x 6) ( x 1) C
2
x 5x
6 7 x 6 7 x 1 7 7 7
12. 2x 1 A B 2x 1 A( x 3) B( x 4); x 3 B 7 7; x 4 A 9 9;
2
x 7x
12 x 4 x 3 1 1
2x 1 dx 9 dx ( 4)
2 4 7 dx
3
9ln 4 7 ln 3 ln x
7
7 12 x x
x x C C
x x ( x 3)
9
13.
2
y
y A B y A( y 1) B( y 3); y 1 B 1 1 ; y 3 A 3 ;
2 y 3 y 3 y 1 4 4 4
8 y dy 3
8 dy 1
8 dy 3 1
8
ln y 3 ln y 1 3 ln 5 1 ln 9 3 ln1 1 ln 5
4
2
y 2y
3 4 4 y 3 4 4 y 1 4 4 4 4 4 4 4
1 1 ln 15
ln 5 ln 3
2 2 2
y 4
14. A B y 4 A( y 1) By;
y 0 A 4; y 1 B 3 3;
2
y y y y 1
1
1 y 4 1 1 1
1 3
1 2 4 dy
1 2 3 dy
1 2 1 4ln 3ln 1 1 2
(4ln1 3ln 2) 4ln 2 3ln 2
dy 2
y y y y
y y
ln 1 ln 1 ln 27 ln 27 1 16 ln 27
8 16 8 8 8 4
15. 1 A B C 1 A( t 2)( t 1) Bt( t 1) Ct( t 2); t 0 A 1 ; t 2 B 1 ;
3 2
t t 2t
t t 2 t 1 2 6
t 1 C 1 1 1 1 1 1 1
3 ; dt dt dt dt
3 2
2 2 t 6 t 2 3 t 1 2 ln | t | 6 ln | t 2| 3
ln | t 1| C
t t t
16. x 3 A B C 1 ( x 3) A( x 2)( x 2) Bx( x 2) Cx( x 2); x 0 A 3 ; x 2 B 1 ;
3
2x
8x
x x 2 x 2 2 8 16
x 2 C 5 3 3 1 5 3 1
5
16 ; x dx dx dx dx
3
2 8 8 16 2 16 2 8 ln x
x x x
16 ln x 2 16
ln x 2 C
x x
5
1
16 ln x x
6
( 2) ( 2)
x
C
17.
3
x ( x 2) 3x
2 (after long division); 3x 2 A B
2 2
x 2x 1 ( x 1)
( x 1) 2 x 1 ( x 1)
2
1 3 1 1 1
A 3, A B 2 A 3, B 1; x dx
0 2 1 0 ( x 2) dx 3 dx dx
x x 0 x 1 0 ( x 1)
2 1
2 2
x
1 1 1
2 2 x 3ln x 1 x 1
0
2 2 3ln 2 2
(1) 3ln 2 2
3x 2 A( x 1) B Ax ( A B)
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