Thomas Calculus 13th [Solutions]

588 Chapter 8 Techniques of Integration
18.
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
A 3, A B 2 A 3, B 1;
2 0
0 3 0 0 0
x dx ( x 2) dx 3 dx dx
1 x 2x 1 1 1 x 1 1 ( x 1)
2 2
x
1 1 1 1
2 2 x 3ln x 1 x 1 0 0 3ln1 1
( 1) 2 2 3ln 2 ( 2)
2 3ln 2
3x 2 A( x 1) B Ax ( A B)
1 A B C D
( x 1) x 1 x 1 ( x 1) ( x 1)
19.
2 2 2 2
2 2 2 2
1 A( x 1)( x 1) B( x 1)( x 1) C( x 1) D( x 1) ;
x 1 C 1 ;
1
3
x 1 D ; coefficient of x A B A B 0; constant A B C D
4
4
A B C D 1 A B 1 ; thus, 1 1
2
A 4 B 4 ;
dx 1 dx 1 dx 1 dx 1 dx 1 ln x 1 x C
2
2 4 x 1 4 x 1 4 2 4 2
( 1) ( 1) 4 x 1 2
x 1
x x 2 x 1
20.
2
x
( x
2
1) x 2x
1
A B C
x 1 x 1 ( x 1)
2
2 2
x A( x 1) B( x 1)( x 1) C( x 1); x 1 C 1 ;
2
x 1 A 1 ; coefficient of
4
2
x A B A B 1 B 3 ; x dx
x x x
4 2
( 1) 2 1
2
ln ( x 1)( x 1)
1 dx 3 dx 1 dx 1 ln x 1 3 ln x 1 1 C 1 C
4 x 1 4 x 1 2 2
( x 1) 4 4 2( x 1) 4 2( x 1)
3
21. 1 A Bx C
2
1 A x 1 ( Bx C)( x 1); x 1 A 1 ; coefficient of
2 1 2
( x 1) x 1 x x 1
2
2
x A B
A B 0 B 1 ; constant 1
1
A C A C 1 C ; dx 1
1
dx 1
1 ( x 1)
dx
2
2 0 2 2 0 1 2 0
2
( x 1) x 1 x x 1
2 1
1
1 1 1 1 1 1 1 1 1 1 1
ln| x 1| ln x 1 tan x ln 2 ln 2 tan 1 ln1 ln1 tan 0
2 4 2 0 2 4 2 2 4 2
1 1 ( 2ln 2)
ln 2
4 2 4 8
22.
2
A
3 2
3t t 4 Bt C 2 2
2
3t t 4 A t 1 ( Bt C) t; t 0 A 4; coefficient of t A B A B
t t t t 1
2
B 1; coefficient of
1; 3
3 4
3 3 ( 1)
1 4
t
t C C t t dt dt
dt
3
1 1
2
t 1 t t 1
4ln
3
1 2 1 1 1 1
1
t
2 ln t 1 tan t 4ln 3 1
2 ln 4 tan 3 4ln1 2
ln 2 tan 1
2ln 3 ln 2 1 ln 2 2ln 3 1 ln 2 ln 9
3 2 4 2 12 2 12
3
23.
2
y 2y 1 Ay B Cy D 2 2 3 2
y 2y 1 ( Ay B) y 1 Cy D Ay By ( A C) y ( B D)
2 2 2
2 1 2
y 1 y y 1
A 0, B 1; A C 2 C 2; B D 1 D 0;
2
y 2 y 1 1
y
dy dy 2 dy
2 2 2
2 1
2
y 1 y
y 1
tan
1 y 1
2
y 1
C
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