Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Please solve only 9 ,21
![2:13 O
O X all 97%I
View embed
IMG_5797.png
484
CHAPTER 7 Techniques of integration
Using Formula I and solving for the required integral, we get
j sec'r de - (sec x tan x + In |sec x + tan x|) + c
Integrals such as the one in the preceding example may seem very special but they
occur frequently in applications of integration, as we will see in Chapter 8. Integrals of
the form cot"x ese"x dx can be found by similar methods because of the identity
1+ cot'x = csce'x.
Finally, we can make use of another set of trigonometric identities:
2 To evaluate the integrals (a) sin mx cos Rx dx, (b) sin mx sin nx dx, or
(c) cos mx cos Rx da, use the corresponding identity:
(a) sin A cos B = (sin(A - B) + sintA + B)]
These product identities are discussed
in Appendix D.
(b) sin A sin B - [cos(A - B) - cos(A + B)]
(c) cos A cos B - [cos(A - B) + cos(A + B)]
EXAMPLE 9 Evaluate sin 4x cos 5x dx.
SOLUTION This integral could be evaluated using integration by parts, but it's easier to
use the identity in Equation 2(a) as follows:
S sin 4x cos Se de - j štsin(-x) + sin 94] dz
x= | [sin(-x) + sin 9x] dx
=|(-sin x + sin 9x) dx
= {(cos x - cos 9x) + c
%3D
7.2 EXERCISES
1-49 Evaluate the integral.
15. ( cot x cos'x dx
16. tan'x cos' dx
1. j sin's cos'r de
2. j sin'o cos'o do
17. sin's sin 2x dx
18. ( sin x confla) de
3. sin'e cos'e de
4. [" sin's de
19. r sin'r dr
20. (x sin'x da
5. sin'(2) cos'(2) de
r cos' (') dt
6.
21. tan x sec'x dx
22. ( tan'e sec'o do
7. "cos'e de
8. " sin (jo) de
23. tan's da
24. (tan's + tan'x) dx
9. cos'(2r) dr
10. sin'r cos't dt
25. tan'x sec'x da
26. sece tan'e de
11. " sin's cos'r da
sin's cos'x de
12. "(2 - sin' de
27. tan'k sec x dx
28. tan'x sec'x dx
13. veos sin'e de
14. sin'1/)
di
29. tan's sec'x dx
30. tan's di
Dipyig S Cmpe leaming AR ked My thepd di wde w. Dknrigh pety](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03fd5f25-eded-4046-9c65-9e89b5e28b89%2F5f52b8d3-80e9-41d2-a5ab-49a2c4fa1c88%2F537f8la_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2:13 O
O X all 97%I
View embed
IMG_5797.png
484
CHAPTER 7 Techniques of integration
Using Formula I and solving for the required integral, we get
j sec'r de - (sec x tan x + In |sec x + tan x|) + c
Integrals such as the one in the preceding example may seem very special but they
occur frequently in applications of integration, as we will see in Chapter 8. Integrals of
the form cot"x ese"x dx can be found by similar methods because of the identity
1+ cot'x = csce'x.
Finally, we can make use of another set of trigonometric identities:
2 To evaluate the integrals (a) sin mx cos Rx dx, (b) sin mx sin nx dx, or
(c) cos mx cos Rx da, use the corresponding identity:
(a) sin A cos B = (sin(A - B) + sintA + B)]
These product identities are discussed
in Appendix D.
(b) sin A sin B - [cos(A - B) - cos(A + B)]
(c) cos A cos B - [cos(A - B) + cos(A + B)]
EXAMPLE 9 Evaluate sin 4x cos 5x dx.
SOLUTION This integral could be evaluated using integration by parts, but it's easier to
use the identity in Equation 2(a) as follows:
S sin 4x cos Se de - j štsin(-x) + sin 94] dz
x= | [sin(-x) + sin 9x] dx
=|(-sin x + sin 9x) dx
= {(cos x - cos 9x) + c
%3D
7.2 EXERCISES
1-49 Evaluate the integral.
15. ( cot x cos'x dx
16. tan'x cos' dx
1. j sin's cos'r de
2. j sin'o cos'o do
17. sin's sin 2x dx
18. ( sin x confla) de
3. sin'e cos'e de
4. [" sin's de
19. r sin'r dr
20. (x sin'x da
5. sin'(2) cos'(2) de
r cos' (') dt
6.
21. tan x sec'x dx
22. ( tan'e sec'o do
7. "cos'e de
8. " sin (jo) de
23. tan's da
24. (tan's + tan'x) dx
9. cos'(2r) dr
10. sin'r cos't dt
25. tan'x sec'x da
26. sece tan'e de
11. " sin's cos'r da
sin's cos'x de
12. "(2 - sin' de
27. tan'k sec x dx
28. tan'x sec'x dx
13. veos sin'e de
14. sin'1/)
di
29. tan's sec'x dx
30. tan's di
Dipyig S Cmpe leaming AR ked My thepd di wde w. Dknrigh pety
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