491
SECTION 7.3 Trigonometric Substitution
We now substitute u − 2 sin , giving du − 2 cos d and s4 2 u 2 − 2 cos , so
Figure 5 shows the graphs of the integrand in Example 7 and its indefinite
integral (with C − 0). Which is which?
y
3
x
2 sin 2 1
dx − y
2 cos d
2
2 cos s3 2 2x 2 x
−
2
_4
y s2 sin 2 1d d
− 22 cos 2 1 C
− 2s4 2 u 2 2 sin21
_5
SD
u
2
FIGURE 5
− 2s3 2 2x 2 x 2 2 sin21
1–3 Evaluate the integral using the indicated trigonometric
substitution. Sketch and label the associated right triangle.
1.
2.
3.
dx
y
x − 2 sin x s4 2 x 2
2
x3
y
sx 2 1 4
dx
x − 2 tan x − 2 sec 4.
5.
7.
y
s9 2 x 2
y
y
3
y
1y2
0
9.
2
11.
0
13.
15.
a
0
y
a.0
dx
sx 2 2 1d3y2
x
3
0
s36 2 x 2
21.
y
8.
y
10.
y
2y3
y
2
dt
x s1 2 4x dx
2
0
s4 1 t 2
y
1
x 2 sa 2 2 x 2 dx
12.
14.
y
x11
2
18.
y
s1 1 x 2
dx
x
20.
y
22.
y
1
y
1
x2
0.6
s9 2 25x 2
dx
dx
24.
sx 1 2x 1 5
25.
yx
27.
y sx
29.
y x s1 2 x
2
s1 1 x 2
0
s3 1 2x 2 x 2 dx
2
x
0
2
1 2x dx
Q
dx
fsaxd2 2 b 2 g 3y2
dx
sx 2 7
2
1C
dx
sx 2 1 1 dx
sx 2 x 2 dx
x2
26.
y s3 1 4x 2 4x
28.
y sx
30.
y
2
2 3y2
d
dx
x2 1 1
dx
2 2x 1 2d2
4
dx
cos t
y2
0
s1 1 sin 2 t
dt
t 2st 2 2 16
0
16.
y
S D
dx
dt
0
sx 2 2 9
dx
x3
y
y
6.
dx
,
sa 2 1 x 2 d3y2
a
y
dx
sx 2 2 1
dx
x4
y
19.
23.
4–30 Evaluate the integral.
x2
x
y
0
sx 2 2 4
dx
x
y
17.
1C
s4 2 9x 2 dx
dx
sx 2 1 1d2
2y3
s2y3
dx
x 5s9x 2 2 1
31. (a) Use trigonometric substitution to show that
y
dx
sx 1 a 2
2
− ln ( x 1 sx 2 1 a 2 ) 1 C
(b) Use the hyperbolic substitution x − a sinh t to show that
y
dx
sx 1 a
2
2
SD
− sinh21
x
a
1C
These formulas are connected by Formula 3.11.3.
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492
CHAPTER 7 Techniques of Integration
(b) Use the figure to give trigonometric interpretations
of both terms on the right side of the equation in part (a).
32. Evaluate
x2
dx
sx 1 a 2 d3y2
y
2
y
a
a@-t@
y=œ„„„„„
(a) by trigonometric substitution.
(b) by the hyperbolic substitution x − a sinh t.
¨
¨
33. Find the average value of f sxd − sx 2 1yx, 1 < x < 7.
2
34. Find the area of the region bounded by the hyperbola
9x 2 2 4y 2 − 36 and the line x − 3.
35. Prove the formula A − 12 r 2 for the area of a sector of a
circle with radius r and central angle . [Hint: Assume
0 , , y2 and place the center of the circle at the
origin so it has the equation x 2 1 y 2 − r 2. Then A is the
sum of the area of the triangle POQ and the area of the
region PQR in the figure.]
y
0
x
40. The parabola y − 12 x 2 divides the disk x 2 1 y 2 < 8 into two
parts. Find the areas of both parts.
41. A torus is generated by rotating the circle x 2 1 s y 2 Rd2 − r 2
about the x-axis. Find the volume enclosed by the torus.
42. A charged rod of length L produces an electric field at point
Psa, bd given by
EsPd − y
L2a
2a
P
t
b
dx
4«0 sx 2 1 b 2 d3y2
where is the charge density per unit length on the rod and
«0 is the free space permittivity (see the figure). Evaluate the
integral to determine an expression for the electric field EsPd.
¨
O
Q
R
x
y
P (a, b)
; 36. Evaluate the integral
y
0
L
x
dx
x sx 2 2 2
4
Graph the integrand and its indefinite integral on the
same screen and check that your answer is reasonable.
37. Find the volume of the solid obtained by rotating
about the x-axis the region enclosed by the curves
y − 9ysx 2 1 9d, y − 0, x − 0, and x − 3.
43. Find the area of the crescent-shaped region (called a lune)
bounded by arcs of circles with radii r and R. (See the figure.)
r
R
38. Find the volume of the solid obtained by rotating
about the line x − 1 the region under the curve
y − x s1 2 x 2 , 0 < x < 1.
39. (a) Use trigonometric substitution to verify that
y
x
0
sa 2 2 t 2 dt − 21 a 2 sin21sxyad 1 12 x sa 2 2 x 2
44. A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are
vertical. If the depth of the water is 7 ft, what percentage of
the total capacity is being used?
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.