SalasSV_08_02_ex - EXERCISES 8.2 Calculate the integral. 1....

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Unformatted text preview: EXERCISES 8.2 Calculate the integral. 1. ± xe − x dx . 2. ± 2 x 2 x dx . 3. ± x 2 e − x 3 dx . 4. ± x ln x 2 dx . 5. ± 1 x 2 e − x dx . 6. ± x 3 e − x 2 dx . 7. ± x 2 √ 1 − x dx . 8. ± dx x (ln x ) 3 . 9. ± e 2 1 x ln √ x dx . 10. ± 3 x √ x + 1 dx . 11. ± ln( x + 1) √ x + 1 dx . 12. ± x 2 ( e x − 1) dx . 13. ± (ln x ) 2 dx . 14. ± x ( x + 5) − 14 dx . 15. ± x 3 3 x dx . 16. ± √ x ln x dx . 17. ± x ( x + 5) 14 dx . 18. ± (2 x + x 2 ) 2 dx . 19. ± 1 / 2 x cos π x dx . 20. ± π/ 2 x 2 sin x dx . 21. ± x 2 ( x + 1) 9 dx . 22. ± x 2 (2 x − 1) − 7 dx . 23. ± e x sin x dx . 24. ± ( e x + 2 x ) 2 dx . 25. ± 1 ln (1 + x 2 ) dx . 26. ± x ln ( x + 1) dx . 27. ± x n ln x dx ( n ±= − 1). 28. ± e 3 x cos 2 x dx . 29. ± x 3 sin x 2 dx . 30. ± x 3 sin x dx . 31. ± 1 / 4 sin − 1 2 x dx . 32. ± sin − 1 2 x √ 1 − 4 x 2 dx . 33. ± 1 x tan − 1 x 2 dx . 34. ± cos √ x dx . HINT: Let u = √ x . 35. ± x 2 cosh 2 x dx . 36. ± 1 − 1 x sinh (2 x 2 ) dx . 37. ± 1 x sin − 1 ( ln x ) dx . 38. ± cos ( ln x ) dx . HINT: Integrate by parts twice. 39. ± sin (ln x ) dx . 40. ± 2 e 1 x 2 (ln x ) 2 dx . 41. Derive Formula (8.2.4): ± ln x dx = x ln x − x + C . 42. Derive Formula (8.2.6): ± tan − 1 x dx = x tan − 1 x − 1 2 ln (1 + x 2 ) + C . Derive the integration formula. 43. ± x k ln x dx = x k + 1 k + 1 ln x − x k + 1 ( k + 1) 2 + C ( k ±= − 1). 44. ± e ax cos bx dx = e ax ( a cos bx + b sin bx ) a 2 + b 2 + C . 458 ± CHAPTER 8 TECHNIQUES OF INTEGRATION 45. ± e ax sin bx dx = e ax ( a sin bx − b cos bx ) a 2 + b 2 + C . 46. What happens if you use integration by parts to evalu-ate ² e ax cosh ax dx ? Evaluate this integral by some other method. 47. If f ( x ) = x sin x on [0, π ], then f (0) = f ( π ) = 0 and f ( x ) > 0 for x ∈ (0, π ). Find the area of the region bounded by the graph of f and the x-axis. 48. If g ( x ) = x cos ( x / 2) on [0, π ], then g (0) = g ( π ) = 0 and g ( x ) > 0 for x ∈ (0, π ). Find the area of the region bounded by the graph of g and the x-axis. In Exercises 49 and 50, ±nd the area of the region bounded by the graph of f and the x-axis. 49. f ( x ) = sin − 1 x , x ∈ [0, 1 2 ]. 50. f ( x ) = xe − 2 x , x ∈ [0, 2]. 51. Let ± be the region bounded by the graph of f ( x ) = ln x and the x-axis between x = 1 and x = e . (a) Find the area of ± . (b) Find the centroid of ± . (c) Find the volume of the solids generated by revolving ± about each of the coordinate axes....
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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SalasSV_08_02_ex - EXERCISES 8.2 Calculate the integral. 1....

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