Thomas Calculus 13th [Solutions]

594 Chapter 8 Techniques of Integration
3 1 1
3 3
57. A tan x dx x tan x x dx
0 0
2
0 1 x
3
3 1 2
3
ln x 1 ln 2;
3 2
0
3
3 1 2 1
3 3 2
x 1 x tan x dx 1 1 x tan x 1 x dx
A 0 A 2 2 0
2
0 1 x
3
1 1 1 1 3 1 2 3
x tan x
1.10
A 2 2
0
A 2 2 6 A 3 2
58.
5 2
5 5 5 5
A 4x 13x 9 125
3 dx 3 dx dx 2 dx 3ln ln 3 2ln 1 ln ;
3 2
x 2x 3x
3 x 3 x 3 3 x 1 x x x 3 9
2
5 x 4x 13x
9
1 1 5 5 5
x dx 4x
3 dx 2 dx 1 (8 11ln 2 3ln 6) 3.90
A 3
3 2
x 2x 3x
A 3 3 x 3 3 x 1 A
59. (a) dx kx( N x) dx k dt 1 dx 1 dx k dt 1 ln x kt C;
dt x( N x)
N x N N x N N x
(b)
k 1
250 , N 1000, t 0 and x 2 1 ln 2 C 1 ln x t 1 ln 1
1000 998 1000 1000 x 250 1000 499
4 4 4 4
ln 499x 499 t t t t
4t x e 499 x e (1000 x) 499 e x 1000e x 1000e
1000 x 1000 x 4t
499 e
4t
1 1000
4t 4t 4t
x N 500 500 e 500 499 500e 1000e e 499 t 1 ln 499 1.55days
2 4t
499 e
4
4t
60.
dx k( a x)( b x)
dx k dt
dt
( a x)( b x)
(a) a b: dx k dt 1 ; 0
2
( a x)
a x
kt C t and x 0 1 C 1 kt 1
a a x a
1 1 a x a x a a a kt
a x a akt 1 akt 1 akt 1
(b) a b: dx k dt 1 dx 1 dx k dt 1 ln b x
( a x)( b x)
b a a x b a b x b a a x
kt C;
t 0 and
( )
1
0 1
b a kt
ab e
x ln b C ln b x ( b a) kt ln b b x b e x
b a a a x a a x a a be
2
( b a)
kt
( b a)
kt
8.6 INTEGRAL TABLES AND COMPUTER ALGEBRA SYSTEMS
1.
dx 2 tan 1 x 3 C
x x 3 3 3
(We used FORMULA 13(a) with a 1, b 3)
2.
dx 1 x 4 4 1 x 4 2
ln
C ln C
x x 4 4 x 4 4 2 x 4 2
(We used FORMULA 13(b) with a 1, b 4 )
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