Thomas Calculus 13th [Solutions]

Section 8.4 Trigonometric Substitutions 579
16.
2 2
2 2 2
x 2 tan , , dx 2 sec d , 4 x 4sec ;
2 2
2 4 tan 2sec
x dx
2 2
4 x
4sec
2 tan 2 C x 2 tan
x
C
d
1
2 2 2
2 tan d 2 sec 1 d 2 sec d 2 d
2
2 d 2 2
2 2 cos
cos
17. x 2 tan , , dx , x 4 ;
2
3
x dx
x
2
4
3
3
2
8 tan cos d sin d cos 1 sin
8 8 ;
2 4 4
cos cos cos
d
[ t cos , dt sin d ]
2 3
2 x 4
4
8 t 1 dt 8 1 1 dt 1 1
sec
x
8 8 sec 8
4 2 4 t
C 3
t t t 3t
3 C 2 8 3
C
2
3 2
1 2 1 2 2
x 4 4 x 4 C x 8 x 4 C
3 3
2
3 2
18.
2 2
x tan , , dx sec d , x 1 sec ;
2 2
x
dx
2 2
x
1
2 2
sec d cos d 1
x 1
C
2 2
tan sec sin sin
x
C
19.
2
w 2 sin , , dw 2 cos d , 4 w 2 cos ;
2 2
8 dw 8 2 cos d 2 4
2 d
w
2 cot C
2 2
2 2
w 4 w 4 sin 2 cos sin
w
2
C
20.
2
w 3 sin , , dw 3 cos d , 9 w 3 cos ;
2 2
2 2
9 w 3cos 3cos d 2 2
dw cot d 1 sin d csc 1 d
2 2 2
w
9 sin sin
cot
C
2
9 w
w
sin
1
w
3
C
21.
x
1
1
dx
x
Multiply the integrand by
1
1
x
.
x
x 1 x 1
dx dx where 1 x 1
1 x
2
1 x
2
x sin , dx cos d , so that cos 0 and 1 x cos .
2 2
x
1
1 sin
dx
2
x cos
1 cos d
sin 1 d cos
1
sin x 1
2
x C
C
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