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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
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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2:12 O ao O X all 97%I View embed IMG_5795.png 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
Transcribed Image Text:2:12 O ao O X all 97%I View embed IMG_5795.png 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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