11. 13.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve only 11,13
![6:12 a ao O
b •
O X O G ull 65%
View embed
IMG_5801.png
534
CHAPTER 7 Techniques of Integration
Therefore, taking f(x) = e* and g(x) = e" in the Comparison Theorem, we see that
jedx is convergent. It follows that e"dx is convergent.
Table 1
In Example 9 we showed that e dx is convergent without computing its value. In
Exercise 72 we indicate how to show that its value is approximately (0.8862. In probability
theory it is important to know the exact value of this improper integral, as we will see in
Section 8.5; using the methods of multivariable calculus it can be shown that the exact
value is T /2. Table I illustrates the definition of an improper integral by showing how
0,7468241328
2.
0,8862073483
0.8862269118
the (computer-generated) values of , e"dx approach r /2 as t becomes large. In fact,
these values converge quite quickly because e 0 very rapidly as x ,
5
0.8862269255
0.8862269255
EXAMPLE 10 The integral dx is divergent by the Comparison Theorem
because
Table 2
[I +e"Va]dx
0,8636306042
1.8276735512
10
2.5219648704
and (1/x) dx is divergent by Example I [or by (2) with p = 1.
100
4.8245541204
1000
7.1271392134
Table 2 illustrates the divergence of the integral in Example 10. It appears that the
values are not approaching any fixed number.
10000
9.4297243064
7.8 EXERCISES
1. Explain why each of the following integrals is improper.
10. ". 2" de
(a)
(b) Ads
11.
13. . "de
12. - 3y') dy
dx
(d) (" cot x da
14. dk
2. Which of the following integrals are improper? Why?
(a) "tan x dx
tan a
(b) (tan x da
16. sind e do
15. sin'a da
(e) f.
da
(d) e"ds
17.
dv
18. --
dx
+ 2
3
3. Find the area under the curve y- 1/x' from x-I to -
and evaluate it for r- 10, 100, and 1000. Then find the total
19. (" ze" dz
20. ve dy
area under this curve for x 1.
H 4. (a) Graph the functions f(x) - 1/x" and g(x) - 1/1
* in
21. da
In x
22. ndx
In x
the viewing rectangles (0, 10] by [o, 11 and [0, 100)
by [0, 1].
(b) Find the areas under the graphs of f and g from x-1
to-rand evaluate for r 10, 100, 10', 10, 10".
and 10
(c) Find the total area under each curve for x> I, if it exists.
23. .d:
24.
dx
x(In x)
+4
da
25. dy
27. [a
20. ["
31. f', da
26.
5-40 Determine whether each integral is convergent or divergent.
Evaluate those that are convergent
28. du
J. 5-
dx
dx
5. -2 de
30. + 1*
6.
dx
29.
dx
7. .3-4
da
8.
(2x + 7 N
32. -
dx
dx
DgsCnge leam R d May da
tpty
e hhup](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03fd5f25-eded-4046-9c65-9e89b5e28b89%2Fcc552f48-b6a3-426e-8073-bda8b2d9c515%2Fmpdpvfc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6:12 a ao O
b •
O X O G ull 65%
View embed
IMG_5801.png
534
CHAPTER 7 Techniques of Integration
Therefore, taking f(x) = e* and g(x) = e" in the Comparison Theorem, we see that
jedx is convergent. It follows that e"dx is convergent.
Table 1
In Example 9 we showed that e dx is convergent without computing its value. In
Exercise 72 we indicate how to show that its value is approximately (0.8862. In probability
theory it is important to know the exact value of this improper integral, as we will see in
Section 8.5; using the methods of multivariable calculus it can be shown that the exact
value is T /2. Table I illustrates the definition of an improper integral by showing how
0,7468241328
2.
0,8862073483
0.8862269118
the (computer-generated) values of , e"dx approach r /2 as t becomes large. In fact,
these values converge quite quickly because e 0 very rapidly as x ,
5
0.8862269255
0.8862269255
EXAMPLE 10 The integral dx is divergent by the Comparison Theorem
because
Table 2
[I +e"Va]dx
0,8636306042
1.8276735512
10
2.5219648704
and (1/x) dx is divergent by Example I [or by (2) with p = 1.
100
4.8245541204
1000
7.1271392134
Table 2 illustrates the divergence of the integral in Example 10. It appears that the
values are not approaching any fixed number.
10000
9.4297243064
7.8 EXERCISES
1. Explain why each of the following integrals is improper.
10. ". 2" de
(a)
(b) Ads
11.
13. . "de
12. - 3y') dy
dx
(d) (" cot x da
14. dk
2. Which of the following integrals are improper? Why?
(a) "tan x dx
tan a
(b) (tan x da
16. sind e do
15. sin'a da
(e) f.
da
(d) e"ds
17.
dv
18. --
dx
+ 2
3
3. Find the area under the curve y- 1/x' from x-I to -
and evaluate it for r- 10, 100, and 1000. Then find the total
19. (" ze" dz
20. ve dy
area under this curve for x 1.
H 4. (a) Graph the functions f(x) - 1/x" and g(x) - 1/1
* in
21. da
In x
22. ndx
In x
the viewing rectangles (0, 10] by [o, 11 and [0, 100)
by [0, 1].
(b) Find the areas under the graphs of f and g from x-1
to-rand evaluate for r 10, 100, 10', 10, 10".
and 10
(c) Find the total area under each curve for x> I, if it exists.
23. .d:
24.
dx
x(In x)
+4
da
25. dy
27. [a
20. ["
31. f', da
26.
5-40 Determine whether each integral is convergent or divergent.
Evaluate those that are convergent
28. du
J. 5-
dx
dx
5. -2 de
30. + 1*
6.
dx
29.
dx
7. .3-4
da
8.
(2x + 7 N
32. -
dx
dx
DgsCnge leam R d May da
tpty
e hhup
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