SalasSV_08_02_ex - 8.2 INTEGRATION BY PARTS 457 and sin1 x...

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EXERCISES 8.2 Calculate the integral. 1. xe x dx . 2. 2 0 x 2 x dx . 3. x 2 e x 3 dx . 4. x ln x 2 dx . 5. 1 0 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 0 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 0 x cos π x dx . 20. π/ 2 0 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 0 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 0 sin 1 2 x dx . 32. sin 1 2 x 1 4 x 2 dx . 33. 1 0 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 .
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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 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, find 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. 52. Let f ( x ) = ln x x , x [1, 2 e ].
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