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504
CHAPTER 7
TECHNIQUES OF INTEGRATION
It’s clear that both systems must have expanded 共x 2 ⫹ 5兲8 by the Binomial Theorem and
then integrated each term.
If we integrate by hand instead, using the substitution u 苷 x 2 ⫹ 5, we get
Derive and the TI-89 and TI-92 also give
this answer.
y x共x
2
⫹ 5兲8 dx 苷
1
18
共x 2 ⫹ 5兲9 ⫹ C
For most purposes, this is a more convenient form of the answer.
EXAMPLE 7 Use a CAS to find
5
2
y sin x cos x dx.
SOLUTION In Example 2 in Section 7.2 we found that
y sin x cos x dx 苷 ⫺
5
1
1
3
2
cos 3x ⫹ 25 cos 5x ⫺ 17 cos7x ⫹ C
Derive and Maple report the answer
⫺ 17 sin 4x cos 3x ⫺ 354 sin 2x cos 3x ⫺
8
105
cos 3x
whereas Mathematica produces
⫺ 645 cos x ⫺
1
192
cos 3x ⫹
3
320
cos 5x ⫺
1
448
cos 7x
We suspect that there are trigonometric identities which show that these three answers
are equivalent. Indeed, if we ask Derive, Maple, and Mathematica to simplify their
expressions using trigonometric identities, they ultimately produce the same form of the
answer as in Equation 1.
Exercises
7.6
1– 4 Use the indicated entry in the Table of Integrals on the
Reference Pages to evaluate the integral.
1.
y
␲兾2
0
2.
y
1
0
3.
y
2
1
4.
y
1
0
cos 5x cos 2x dx ; entry 80
y
0
7.
y
cos x
dx
sin 2 x ⫺ 9
CAS Computer algebra system required
y
2
0
8.
y
t 2e⫺t dt
y
15.
ye
17.
y y s6 ⫹ 4y ⫺ 4y
19.
y sin x cos x ln共sin x兲 dx
x 2s4 ⫺ x 2 dx
21.
y 3⫺e
ln (1 ⫹ sx
sx
23.
tan3共␲ x兾6兲 dx ; entry 69
6.
dx
s4x 2 ⫹ 9
13.
s4x 2 ⫺ 3 dx ; entry 39
arctan 2x dx
0
2
y
sx ⫺ x 2 dx ; entry 113
␲兾8
yx
11.
5–32 Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.
5.
9.
) dx
⫺1
tan 3共1兾z兲
dz
z2
2x
arctan共e x 兲 dx
2
ex
2x
dx
5
y sec x dx
1. Homework Hints available at stewartcalculus.com
2
dy
s2y 2 ⫺ 3
dy
y2
10.
y
12.
yx
14.
y sin
16.
y x sin共x
18.
y 2x
20.
y s5 ⫺ sin ␪
22.
y
csch共x 3 ⫹ 1兲 dx
⫺1
3
sx dx
2
兲 cos共3x 2 兲 dx
dx
⫺ 3x 2
sin 2␪
2
0
24.
2
d␪
x 3 s4x 2 ⫺ x 4 dx
y sin
6
2x dx
DISCOVERY PROJECT
s4 ⫹ 共ln x兲 2
dx
x
26.
y
cos⫺1 共x ⫺2 兲
dx
x3
28.
29.
y
se 2x ⫺ 1 dx
30.
y
31.
y sx
x 4 dx
10 ⫺ 2
32.
y s9 ⫺ tan ␪
25.
27.
y
y
1
0
x 4e⫺x dx
y 共t ⫹ 1兲st
2
⫺ 2t ⫺ 1 dt
e t sin共␣ t ⫺ 3兲 dt
sec 2␪ tan 2␪
2
d␪
CAS
2
39.
y x sx
41.
y cos
43.
PATTERNS IN INTEGRALS
⫹ 4 dx
dx
x
⫹ 2兲
40.
y e 共3e
x dx
42.
y x s1 ⫺ x
y tan x dx
44.
y
4
2
5
505
x
2
1
3
x
s1 ⫹ s
2
dx
dx
45. (a) Use the table of integrals to evaluate F共x兲 苷
x f 共x兲 dx,
where
f 共x兲 苷
33. The region under the curve y 苷 sin 2 x from 0 to ␲ is rotated
1
x s1 ⫺ x 2
about the x-axis. Find the volume of the resulting solid.
What is the domain of f and F ?
(b) Use a CAS to evaluate F共x兲. What is the domain of the
function F that the CAS produces? Is there a discrepancy
between this domain and the domain of the function F
that you found in part (a)?
34. Find the volume of the solid obtained when the region under
the curve y 苷 arcsin x, x 艌 0, is rotated about the y-axis.
35. Verify Formula 53 in the Table of Integrals (a) by differentia-
tion and (b) by using the substitution t 苷 a ⫹ bu.
36. Verify Formula 31 (a) by differentiation and (b) by substi-
CAS
46. Computer algebra systems sometimes need a helping hand
tuting u 苷 a sin ␪.
CAS
from human beings. Try to evaluate
y 共1 ⫹ ln x兲 s1 ⫹ 共x ln x兲
37– 44 Use a computer algebra system to evaluate the integral.
Compare the answer with the result of using tables. If the
answers are not the same, show that they are equivalent.
37.
4
y sec x dx
DISCOVERY PROJECT
38.
dx
with a computer algebra system. If it doesn’t return an
answer, make a substitution that changes the integral into
one that the CAS can evaluate.
5
y csc x dx
CAS
2
PATTERNS IN INTEGRALS
In this project a computer algebra system is used to investigate indefinite integrals of families of
functions. By observing the patterns that occur in the integrals of several members of the family,
you will first guess, and then prove, a general formula for the integral of any member of the family.
1. (a) Use a computer algebra system to evaluate the following integrals.
1
(i)
y 共x ⫹ 2兲共x ⫹ 3兲 dx
(iii)
y 共x ⫹ 2兲共x ⫺ 5兲 dx
1
1
(ii)
y 共x ⫹ 1兲共x ⫹ 5兲 dx
(iv)
y 共x ⫹ 2兲
1
2
dx
(b) Based on the pattern of your responses in part (a), guess the value of the integral
1
y 共x ⫹ a兲共x ⫹ b兲 dx
if a 苷 b. What if a 苷 b?
(c) Check your guess by asking your CAS to evaluate the integral in part (b). Then prove it
using partial fractions.
CAS Computer algebra system required