megration by parts with the 13. j rese' r de 15. (In a' dx -Va de 17. e" sin 30 a 19. :'e'd: 4. dy (I + 2x 6. (x- 1) sin wx du 23. * cos EX 8. sin Br dt 25. y sinh y dy 10. Í In vi de In R dR R 27.
megration by parts with the 13. j rese' r de 15. (In a' dx -Va de 17. e" sin 30 a 19. :'e'd: 4. dy (I + 2x 6. (x- 1) sin wx du 23. * cos EX 8. sin Br dt 25. y sinh y dy 10. Í In vi de In R dR R 27.
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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Please solve 10 and 13 only
Transcribed Image Text:2:12 O ao
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476
CHAPTER 7 Techniques of integration
SOLUTION Let
u= sin'
de = sin rdr
Then
du - (n- 1) sinx cos x dr
- -cosx
so integration by parts gives
| sin"x dx = -cos x sin'x + (n - 1) | sin"x cos'x dx
Since cos'x- I - sin'x, we have
| sin'x dx = -cos x sin'x + (n – 1) | sinx dx – (n - 1) sin"x dx
As in Example 4, we solve this equation for the desired integral by taking the last term
on the right side to the left side. Thus we have
n( sin'x dx = -cos x sin"x + (n - 1) sin"r dx
| sin's dx = - cos x sink + "- sin" de
cos x sin +
or
The reduction formula (7) is useful because by using it repeatedly we could eventu-
ally express sin"x dx in terms of sin x dx (if n is odd) or (sin xy'dx = | dx (if n is
even).
7.1 EXERCISES
1-2 Evaluate the integral using integration by parts with the
indicated choices of u and de.
13. r cse' dt
14. (x cosh ax dx
1. e" dx; w- r, de -" du
15. (In x dx
-d:
10
2. i inx de: w- Inx, de- a de
17. je" sin 30 do
18. e"cos 20 de
19. :'d:
20. x tan'x de
3-36 Evaluate the integral.
3. Jz cos Sx de
4. j ye dy
22. ( (arcsin x da
(1 + 2x
s. fte" dt
6. ((x - 1) sin x ds
23.x cos EN da
24. ( + De"dx
7. (x + 21) cas x du
8. (' sin ar dt
25. y sinh y dy
26. * In w dw
In R
9. f cos "a de
11. fr'inr dt
10. ( In va de
27. dR
28. " sin 2t dr
12. ( tan 2y dy
29.X sin x cos x dx
30. (aretan(1/a) da
Dyige ConL A M
く
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