Thomas Calculus 13th [Solutions]

Chapter 8 Practice Exercises 637
56. The probability that both systems fail is 0.0148. Since the two systems have the same performance
distribution, the failure probability for a single system is 0.0148 0.121655. The success probability for a
2
single system is 1 0.0148 so the probability that both succeed is 1 0.0148 0.771489. The
probability that one fails and one succeeds is 1 0.0148 0.771489 0.213711. Since the events main fails,
backup succeeds and main succeeds, backup fails have the same probability, the probability that only the
main fails is 0.213711/ 2 0.106856. Thus the probability that the main fails, either along with the backup or
by itself, is 0.0148 0.106856 0.121656.
CHAPTER 8
PRACTICE EXERCISES
1.
dx
x 1
u ln( x 1), du ; dv dx, v x;
x
dx
x 1 x 1
ln( x 1) dx x ln( x 1) dx x ln( x 1) dx x ln( x 1) x ln( x 1) C
( x 1)ln( x 1) x C ( x 1)ln( x 1) ( x 1) C , where C C1 1
1
1
dx 2 3
dv x dx, v
1
x ;
x
3
2 3 3
3 3
ln
1
ln
1 1 x
ln
x
3 3 x 3 9
2. u ln x, du ;
x x dx x x x dx x C
1
3 dx
3. u tan 3 x, du ; dv dx, v x;
2
4.
1 9x
1 1
3x dx
2
1 9x
tan 3x dx x tan 3 x ;
1
x
2
u cos , du ; dv dx, v x;
4
dx
1 x
1
dx x
x
2 2
cos cos ;
x
2
x dx
4
x
2
y
1 9x
2
dy 18x dx
y 4
x
2
dy 2x dx
1 1 dy 1 1 2
6 y
6
x tan 3x x tan (3 x) ln 1 9x C
1 x 1 dy 1 x
2
2 2 y
2
x cos xcos 4 x C
1
x
x
2 2
x cos 2 1
2
C
5.
( x
2
1)
2( x 1)
2
( )
( )
( )
x
e
x
e
x
e
x
e
0
2 x
2
x
( x 1) e dx ( x 1) 2( x 1) 2 e C
6.
sin(1 x)
2
x
( )
cos(1 x)
2x
( )
sin(1 x)
2
( )
cos(1 x)
0
2 2
x sin(1 x) dx x cos(1 x) 2xsin(1 x) 2cos(1 x)
C
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