❙❙❙❙
510
CHAPTER 7 TECHNIQUES OF INTEGRATION
then we know from Part 1 of the Fundamental Theorem of Calculus that
F"!x" ! e x
2
2
Thus, f !x" ! e x has an antiderivative F , but it has been proved that F is not an elementary function. This means that no matter how hard we try, we will never succeed in evalu2
ating x e x dx in terms of the functions we know. (In Chapter 11, however, we will see how
2
to express x e x dx as an infinite series.) The same can be said of the following integrals:
y
ex
dx
x
y sx
3
y sin!x
y
! 1 dx
2
" dx
1
dx
ln x
y cos!e
y
x
" dx
sin x
dx
x
In fact, the majority of elementary functions don’t have elementary antiderivatives. You
may be assured, though, that the integrals in the following exercises are all elementary
functions.
|||| 7.5
1–80
y
3.
y
sin x ! sec x
dx
tan x
0
5.
y
1
7.
y
3
9.
2t
dt
!t # 3"2
2
e
dy
1 ! y2
r 4 ln r dr
x#1
dx
x 2 # 4x ! 5
y
yx
3x 2 # 2
dx
2
# 2x # 8
26.
yx
27.
y cot x ln!sin x" dx
28.
y sin sat dt
29.
y
30.
y %x
31.
y
32.
y
x
dx
x4 ! x2 ! 1
33.
y s3 # 2x # x
34.
y%
12.
y sin x cos!cos x" dx
35.
y
36.
y sin 4x cos 3x dx
14.
y
37.
y
38.
y
16.
y
39.
y
40.
y s4y
41.
y $ tan $ d$
42.
yx
43.
y e s1 ! e
44.
y s1 ! e
#1
45.
yx e
46.
y 1#e
2
47.
yx
48.
y
2.
4.
arctan y
#1
1
25.
Evaluate the integral.
||||
1.
Exercises
y tan $ d$
3
x
dx
s3 # x 4
y
6.
y x csc x cot x dx
8.
y
10.
x#1
dx
x 2 # 4x # 5
4
0
y
11.
y sin $
13.
y !1 # x
15.
y
17.
y x sin x dx
18.
y 1!e
19.
ye
20.
ye
21.
yt e
22.
y x sin
23.
y (1 ! sx ) dx
24.
y ln!x
3
5
cos $ d$
dx
1#2
0
"
2 3#2
x
dx
s1 # x 2
2
x!e x
dx
3 #2t
1
0
dt
8
s1 ! ln x
dx
x ln x
x2
dx
s1 # x 2
s2#2
0
e 2t
3
sx
4t
dt
dx
x dx
# 1" dx
3w # 1
dw
w!2
5
0
$
1
1!x
dx
1#x
2
dx
x 8 sin x dx
#1
%#4
0
cos2$ tan2$ d$
x
dx
1 # x 2 ! s1 # x 2
2
x
x
dx
5 #x 3
dx
x!a
dx
2
! a2
3x 2 # 2
dx
3
# 2x # 8
2
2
#2
%
# 4x dx
s2x # 1
dx
2x ! 3
% #2
#4
% #4
0
2
1 ! 4 cot x
dx
4 # cot x
tan 5$ sec 3$ d$
2
1
dy
# 4y # 3
tan#1x dx
1 ! ex
x
x
dx
dx
x
dx
x4 # a4
❙❙❙❙
SECTION 7.6 INTEGRATION USING TABLES AND COMPUTER ALGEBRA SYSTEMS
49.
1
y x s4x ! 1 dx
1
51.
y x s4x
53.
yx
55.
50.
yx
2
1
dx
s4x ! 1
dx
! 1"
arctanst
dt
st
67.
y
69.
y 1!e
71.
yx
3
1
52.
y x !x
54.
y !x ! sin x" dx
y x ! 4 ! 4 sx ! 1 dx
56.
y sx
x ln x
dx
2
#1
73.
y !x # 2"!x
57.
y x sx ! c dx
58.
yx
ln!1 ! x" dx
75.
y sin x sin 2x sin 3x dx
59.
ye
1
dx
# ex
60.
y x ! sx dx
77.
y 1!x
61.
yx
x4
dx
! 16
62.
y !x ! 1"
79.
y x sin
63.
y sxe
64.
y%
66.
y
65.
2
2
!1
dx
sinh mx dx
1
3
y
3x
10
sx
dx
1
dx
sx ! 1 ! sx
|||| 7.6
4
2
2
1
3
x3
10
%#3
#4
3
2
dx
ln!tan x"
dx
sin x cos x
u3 ! 1
du
u3 # u2
■
e 2x
4
x
dx
x
dx
! 4x 2 ! 3
1
sx
■
2
3
2
! 4"
dx
dx
x cos x dx
■
■
■
■
2
1
68.
y 1 ! 2e
70.
y
72.
y 1 ! st dt
74.
ye
76.
y !x
78.
y sin x ! sec x dx
80.
y sin
x
# e#x
511
dx
ln!x ! 1"
dx
x2
st
3
x
dx
# e #x
2
# bx" sin 2x dx
sec x cos 2x
sin x cos x
dx
4
x ! cos 4 x
■
■
■
■
■
■
2
81. The functions y ! e x and y ! x 2e x don’t have elementary
2
antiderivatives, but y ! !2x 2 ! 1"e x does. Evaluate
2
x !2x 2 ! 1"e x dx.
Integration Using Tables and Computer Algebra Systems
In this section we describe how to use tables and computer algebra systems to integrate
functions that have elementary antiderivatives. You should bear in mind, though, that even
the most powerful computer algebra systems can’t find explicit formulas for the antideriv2
atives of functions like e x or the other functions described at the end of Section 7.5.
Tables of Integrals
Tables of indefinite integrals are very useful when we are confronted by an integral that is
difficult to evaluate by hand and we don’t have access to a computer algebra system. A relatively brief table of 120 integrals, categorized by form, is provided on the Reference Pages
at the back of the book. More extensive tables are available in CRC Standard Mathematical Tables and Formulae, 30th ed. by Daniel Zwillinger (Boca Raton, FL: CRC
Press, 1995) (581 entries) or in Gradshteyn and Ryzhik’s Table of Integrals, Series,
and Products, 6e (New York: Academic Press, 2000), which contains hundreds of pages of
integrals. It should be remembered, however, that integrals do not often occur in exactly
the form listed in a table. Usually we need to use substitution or algebraic manipulation to
transform a given integral into one of the forms in the table.
EXAMPLE 1 The region bounded by the curves y ! arctan x, y ! 0, and x ! 1 is rotated
about the y-axis. Find the volume of the resulting solid.
SOLUTION Using the method of cylindrical shells, we see that the volume is
V ! y 2% x arctan x dx
1
0