MS-Sol-EE-C08

MS-Sol-EE-C08 - CHAPTER 8 INDEFINITE INTEGRATION EXERCISE...

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CHAPTER 8 INDEFINITE INTEGRATION EXERCISE 8.1 Section 8.1 Primitive functions and indefinite integrals (page 318) 1. 6 6 1 x dx d = x 5 dx x 5 = 6 1 x 6 + C 2. dx d (2 x 4 ) = 8 x 3 8 x 3 dx = 2 x 4 + C 3. 2 1 x dx d = 3 2 x - dx x 2 3 - = 2 1 x + C 4. dx d (4 x 2 3 ) = 6 x x 6 dx = 4 x 2 3 + C 5. ) 3 ( 3 x dx d = 3 2 1 x 3 2 1 x dx = 3 3 x + C 6. x x dx d 2 + 3 1 3 = x 2 + 2 dx x 2) + ( 2 = 3 1 x 3 + 2 x + C 7. x e dx d 5 5 1 = e 5 x dx e x 5 = x e 5 5 1 + C 80
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C HAPTER 8 I NDEFINITE I NTEGRATION 8. - - - 2 7 2 7 1 x e dx d x = e –7 x – 4 x - - ) 4 ( 7 x e x dx = – 7 1 e –7 x – 2 x 2 + C 9. [ ] ) 1 ( ln - x dx d = 1 1 - x dx x 1 1 - = ln ( x - 1) + C 10. - - x x dx d ln 3 = 3 x –2 x –1 ) 3 ( 1 2 - - - x x dx = – x 3 – ln x + C 11. 3 dx = 3 x + C 12. - dx 2 = - 2 x + C 13. dx x 6 = 3 x 2 + C 14. 3 4 x dx = x 4 + C 15. - dx x 4 = 3 3 - - x + C 16. x 1 dx = 2 x + C 17. dx e x = e x + C 18. x e 2 dx = 2 1 e 2 x + C 19. - dx e x 3 = x e 3 3 1 - - + C 20. dx x 7 = 7 ln x + C 81
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S ECTION 8.1 P RIMITIVE F UNCTIONS AND I NDEFINITE I NTEGRALS 21. 6 14 x dx = kx 7 + C dx d ( kx 7 ) = 14 x 6 k 7 x 6 = 14 x 6 k 7 = 14 k = 2 22. dx x 3 2 = k 3 5 x + C ) ( 3 5 kx dx d = 3 2 x k 3 5 3 2 x = 3 2 x k 3 5 = 1 k = 5 3 23. 3 ) 1 5 ( - x dx = k (5 x – 1) 4 + C dx d [ k (5 x – 1) 4 ] = (5 x – 1) 3 k 4(5 x – 1) 3 (5) = (5 x – 1) 3 20 k = 1 k = 20 1 24. dx x 7 + 2 = 2 3 7) + 2 ( x k + C d dx [ 2 3 7) + 2 ( x k ] = 7 + 2 x k 2 1 7) + 2 ( 2 3 x (2) = 7 + 2 x 3 k = 1 k = 3 1 25. dx x 2 5 = 4 x k + C 4 x k dx d = 5 2 x 5 ) 4 ( x k - = 5 2 x k ( - 4) = 2 k = - 2 1 82
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HAPTER 8 I NDEFINITE I NTEGRATION 26. - dx e x 4 6 = ke –6 x + C d dx ( ke –6 x ) = 4 e –6 x k ( - 6 e –6 x ) = 4 e –6 x - 6 k = 4 k = - 2 3 27. dy dx = 2 x y = dx x 2 = x 2 + C When x = 3, y = 10. 10 = 3 2 + C C = 1 Thus, y = x 2 + 1. 28. f ( x ) = e x f ( x ) = dx e x = e x + C (0, - 2) is on the curve. - 2 = e 0 + C C = - 3 i.e. f ( x ) = e x - 3 29. dy dx = 2 1 x y = dx x 2 1 = - 1 x + C ( - 1, 3) is on the curve. 3 = ) 1 ( 1 - - + C C = 2 i.e. y = - x 1 + 2 30. (a) 2 1) + ( 2 x dx d = 1 2 2( x + 1) (1) = x + 1 dx x 1) + ( = 2 1) + ( 2 x + C 1 . . . . . . . . . . . . . (1) where C 1 is an arbitrary constant. + x x dx d 2 2 = x + 1 dx x 1) + ( = 2 2 x + x + C 2 . . . . . . . . . . . . . . (2) where C 2 is an arbitrary constant. 83
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This note was uploaded on 10/17/2011 for the course IELM 3010 taught by Professor Fugee during the Winter '11 term at HKUST.

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MS-Sol-EE-C08 - CHAPTER 8 INDEFINITE INTEGRATION EXERCISE...

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