Thomas Calculus 13th [Solutions]

510 Chapter 7 Integrals and Transcendental Functions
4 ln 2log 4 4
4
2 x 2 2 2 2 2
39. dx ln 2 ln x dx ln x dx 1 (ln x) 1 [(ln 4) (ln1) ] 1 (ln 4) 1 (2ln 2)
1 x 1 x ln 2 1 x 2 1 2 2 2
2
2(ln 2)
e 2ln10(log 10 x) e (ln10)(2ln x) 2 2 2
40. 1
e
dx dx [(ln x) ]
1 1 (ln10)
1 (ln e) (ln1) 1
x
x
41.
2
2 log 2
2 ( x 2) 1 1 1 (ln( x 2)) 1 (ln 4) (ln 2)
dx [ln( x 2)] dx
0 x 2 ln 2 0
x 2 ln 2 2 ln 2 2 2
0
2 2
1 4(ln 2) (ln 2) 3 ln 2
ln 2 2 2 2
2 2 2
42.
10
10 log 10
10 (10 x) 10 1 10 (ln(10 x)) 10 (ln100) (ln1)
dx [ln(10 x)]
dx
1/10 x ln10 1/10
10x
ln10 20 ln10 20 2
1/10
10 4(ln10)
ln10 20
2
2ln10
2 2 2
43.
44.
45.
2 9
9 2 2
2log 9
10 ( x 1) 2 1 2 (ln( x 1)) 2 (ln10) (ln1)
dx ln( x 1) dx
ln10
0 x 1 ln10 0
x 1 ln10 2 ln10 2 2
0
2 3
3 2 2
2log 3
2 ( x 1) 2 1 2 (ln( x 1)) 2 (ln 2) (ln1)
dx ln( x 1) dx
ln 2
2 x 1 ln 2 2
x 1 ln 2 2 ln 2 2 2
2
dx ln10 1 (ln10) 1 1 ;
xlog x ln x x dx ln x x
dx u ln x du 1 dx
x
10
(ln10) 1 1 dx (ln10) 1 du (ln10) ln u C (ln10) ln ln x C
ln x x u
46.
dx
dx
2 2
x(log 8 x)
ln x
x
ln 8
2 1 2
2 (ln x) 2 (ln x) (ln 8)
(ln8) dx (ln8) C C
x
1 ln x
47. dy e t sin( e t 2) y e t sin( e t 2) dt;
dt
t t t
let u e 2 du e dt y sin u du cosu C cos( e 2) C;
y(ln 2) = 0
ln 2
t
cos( e 2) C 0 cos(2 2) + C = 0 C = cos 0 = 1; thus, y 1 cos( e 2)
48.
dy 2 2
e t sec ( e t ) y e t sec ( e t ) dt;
dt
t t 2
let
1 t 1 sec 1 tan 1 t
u e du e dt du e dt y u du u C tan( e ) C;
ln 4
y(ln 4) 2 1 tan( e ) C 2 1 tan 1 C 2 1 (1) C 2 C 3 ;
4
thus, 3 1 t
y tan( e )
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