Thomas Calculus 13th [Solutions]

640 Chapter 8 Techniques of Integration
30. (a)
(b)
x dx 1 2x dx
2
4 x C
2 2 2
4 x 4 x
x dx
4
x
2
;
2
2 tan y 2sec y dy
2
[ x 2 tan y] 2 sec y tan y dy 2sec y C 4 x C
2sec y
x dx ( 2 ) 2
31. (a) 1 x dx 1 ln 4 x C
2 2 2
4 x
4 x 2
x dx
2sin 2cos d
2
4 x
(b) ; [ x 2sin ] tan d ln | cos | C ln C
2
2
4 x
4cos
2
1 2
2 ln 4 x C
32. (a)
(b)
t dt 1 8t dt 1 2
4t
1 C
2 8 2
4t
1 4t
1
4
t dt
;
2
4t
1
1 1
d
2
2 2
t
1
sec tan sec
1 2 tan 4 1
t sec
sec d C C
2 tan 4 4 4
x dx
33. ;
2
9
2
u 9 x
du
2x dx
1 du 1 ln | u | C ln 1 C ln 1 C
2 u 2
u 9 x
x 2
34. dx 1 dx 1 dx 1 dx 1 1 1 1 1 2
ln | x | ln | 3 x | ln | 3 x | C ln | x | ln 9 x C
2
x 9 x 9 x 18 3 x 18 3 x 9 18 18 9 18
35. dx 1 dx 1 dx 1 ln | 3 x | 1 ln | 3 x | C 1 ln x 3 C
2
9 x 6 3 x 6 3 x 6 6 6 x 3
36.
9
dx
x
2
;
x
dx
sin
3cos
d
3cos
1
d d C sin x C
3cos 3
37.
5 7
3 4 4 2 4 6
sin x cos x dx cos x 1 cos x sin x dx cos xsin x dx cos xsin x dx cos x cos x C
5 7
38.
5 5 5 4 5 2
2
cos xsin x dx sin x cos x cos x dx sin x 1 sin x cos x dx
6 8 10
5 7 9
sin x cos x dx 2 sin x cos x dx sin x cos x dx sin x 2sin x sin x C
6 8 10
39.
40.
4 2
tan xsec x dx tan 5 x C
5
3 3 2 2 4 2
tan xsec x dx sec x 1 sec x sec x tan x dx sec x sec x tan x dx sec x sec x tan x dx
5 3
sec x sec x C
5 3
41. sin 5 cos6 d 1 sin( ) sin(11 ) d 1 sin( ) d 1 sin(11 ) d 1 cos( ) 1 cos11 C
2 2 2 2 22
1 cos 1 cos11 C
2 22
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