James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

502 Chapter 7 Techniques of Integration
57–58 Evaluate the integral by completing the square and using
Formula 6.
57. y
dx
x 2 2 2x
58. y
2x 1 1
4x 2 1 12x 2 7 dx
59. The German mathematician Karl Weierstrass (1815–1897)
noticed that the substitution t − tansxy2d will convert any
rational function of sin x and cos x into an ordinary rational
function of t.
(a) If t − tansxy2d, 2 , x , , sketch a right triangle or
use trigonometric identities to show that
cosS
2D x 1
−
and sinS x t
−
s1 1 t 2D 2 s1 1 t 2
(b) Show that
cos x − 1 2 t 2
1 1 t 2 and sin x − 2t
1 1 t 2
(c) Show that
dx − 2
1 1 t 2 dt
60–63 Use the substitution in Exercise 59 to transform the integrand
into a rational function of t and then evaluate the integral.
dx
60. y
1 2 cos x
1
61. y
3 sin x 2 4 cos x dx 62. y y2 1
y3 1 1 sin x 2 cos x dx
63. y y2
0
sin 2x
2 1 cos x dx
64–65 Find the area of the region under the given curve from
1 to 2.
64. y − 1
x 3 1 x
65. y − x 2 1 1
3x 2 x 2
66. Find the volume of the resulting solid if the region under the
curve y − 1ysx 2 1 3x 1 2d from x − 0 to x − 1 is rotated
about (a) the x-axis and (b) the y-axis.
67. One method of slowing the growth of an insect population
without using pesticides is to introduce into the population a
number of sterile males that mate with fertile females but produce
no offspring. (The photo shows a screw-worm fly, the
first pest effectively eliminated from a region by this method.)
USDA
CAS
Let P represent the number of female insects in a population
and S the number of sterile males introduced each
generation. Let r be the per capita rate of production of
females by females, provided their chosen mate is not
sterile. Then the female population is related to time t by
t − y
P 1 S
Pfsr 2 1dP 2 Sg dP
Suppose an insect population with 10,000 females grows
at a rate of r − 1.1 and 900 sterile males are added
initially. Evaluate the integral to give an equation relating
the female population to time. (Note that the resulting
equation can’t be solved explicitly for P.)
68. Factor x 4 1 1 as a difference of squares by first adding
and subtracting the same quantity. Use this factorization
to evaluate y 1ysx 4 1 1d dx.
69. (a) Use a computer algebra system to find the partial
fraction decomposition of the function
f sxd −
4x 3 2 27x 2 1 5x 2 32
30x 5 2 13x 4 1 50x 3 2 286x 2 2 299x 2 70
(b) Use part (a) to find y f sxd dx (by hand) and compare
with the result of using the CAS to integrate f
directly. Comment on any discrepancy.
70. (a) Find the partial fraction decomposition of the
function
f sxd −
12x 5 2 7x 3 2 13x 2 1 8
100x 6 2 80x 5 1 116x 4 2 80x 3 1 41x 2 2 20x 1 4
(b) Use part (a) to find y f sxd dx and graph f and its
indefinite integral on the same screen.
(c) Use the graph of f to discover the main features of
the graph of y f sxd dx.
71. The rational number 22 7 has been used as an approximation
to the number since the time of Archimedes. Show
that
y 1
0
x 4 s1 2 xd 4
dx − 22 1 1 x 2 7 2
72. (a) Use integration by parts to show that, for any positive
integer n,
y
dx
sx 2 1 a 2 d n dx −
(b) Use part (a) to evaluate
y
x
2a 2 sn 2 1dsx 2 1 a 2 d n21
1 2n 2 3
2a 2 sn 2 1d
y
dx
sx 2 1 1d and dx
y 2 sx 2 1 1d 3
dx
sx 2 1 a 2 d n21
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