Please #27, 29, 31, 33, 35, 37, 39, 41,45, 57, 61, 69, 77

25. z'e' dz 26. (arcsin x)2 dx (b) Use part (a) to evaluate ] cos x dx. (c) Use parts (a) and (b) to evaluate ( cos x dx. 27 . ( (1 + x ) e 3 dx 28. 0 sin 370 de 55. (a) Use the reduction formula in Example 6 to show that n - 1 (#/2 sin"- 2x dx 29 . ['x 3'dx 30. J (1+ x)2 dx "sin"x dx = where n > 2 is an integer. 31. y sinh y dy 32. (w2 In w du (b) Use part (a) to evaluate " sin'x dx and for sin'x dx. (c) Use part (a) to show that, for odd powers of sine, "s In R 33. R2 - dR 34. 12 sin 2t dt (" sin 2n t x dx = = 2 . 4 . 6 . . .. . 2n 3 .5 . 7 . ... . (2n + 1) 35. x sin x cos x dx 36. arctan(1/x) dx 56. Prove that, for even powers of sine, 37. -M dM 38. 2 (In x)2 3 dx " sin "x dx = 1 . 3 . 5 . ... . (2n - 1) T x 3 2 . 4 . 6 . ... . 2n 2 39 . "sin x In(cos x) dx 40. 57-60 Use integration by parts to prove the reduction formula. dr 57. (In x)"dx = x(Inx)" - n f (Inx)"-'dx 41. cos x sinh x dx 42. [' e'sin(t - s) ds 58. (xhe'dx = x"ex - nfx"-le'dx 43-48 First make a substitution and then use integration by parts to evaluate the integral 59. (tan"x dx = tan" 'x n - 1 - tan " Exdx (n # 1) 43. eva dx 44. cos(In x) dx n - 1 60. [ sec"xax = tan x sec" x 4 " - 2 ( sec"- 2x dx (n # 1) 45. ( 0' cos(0?) de 46. " ecos' sin 2t dt 61. Use Exercise 57 to find f (In x) dx. 47. ( xIn(1 + x) dx 48. arcsin(In x) 62. Use Exercise 58 to find [ xtel dx. X 63-64 Find the area of the region bounded by the given curves. 49-52 Evaluate the indefinite integral. Illustrate, and check that 63. y = x Inx, y = 4 Inx 64 . y = xe , y = xe " your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 65-66 Use a graph to find approximate x-coordinates of the 49. xe -2x dx 50. x 3/2 In x dx points of intersection of the given curves. Then find (approxi- mately) the area of the region bounded by the curves. 51 . (x ' V I + x 2 dx 52 . x2 sin 2x dx 65. y = arcsin(zx), y = 2 -x2 66. y = x In(x + 1), y = 3x - x' 53. (a) Use the reduction formula in Example 6 to show that 67-70 Use the method of cylindrical shells to find the volume [ sin'x dx = * _ sin 2x generated by rotating the region bounded by the curves about the 4 - + c given axis. (b) Use part (a) and the reduction formula to evaluate 67. y = cos( 7x/2), y = 0, 0