Please #69, 73, 75, 79, 81, 83, 89

45. f(x) = 46. f(x) = 63-66 Use series to approximate the definite integral to within V4+ x2 V2 + x the indicated accuracy. 47. f(x) = sin x Hint: Use sin2x = (1 - cos 2x).] 63. x' arctan x dx (four decimal places) x - sin x if x * 0 64. sin(x*) dx (four decimal places) 48. f(x) = if x = 0 65. VI + x dx (lerror) (-1)" n x 4 n! 80. E(-1) "-1 2 4 "=0 n 58. (a) Expand 1/V/1 + x as a power series. (b) Use part (a) to estimate 1/41.1 correct to three decimal x2n+1 x2n+1 81. (-1)" places. 22"+1 ( 2n + 1 ) 82. 2(-1)" - 2 2 + 1 ( 2n + 1 ) ! 59-62 Evaluate the indefinite integral as an infinite series. 83-90 Find the sum of the series. 59. ( Vitx dx 60. x2 sin(x?) dx 83. (-1)" 84. (-1)" TT 20 n! 7 0 62"(2n)! 61. [ cos x -dx 62. arctan(x?) dx 85. E (-1)"-1 3" 86. 3" n 5" 7 0 5" n!(-1)" TT 20+1 87. 96. (a) Show that the function defined by 1=0 42n+ (2n + 1)! e-1/x2 f ( x ) = if x * 0 (In 2)2 (In 2)3 88. 1 - In 2 + + . . . if x = 0 2! 3! 9 27 81 is not equal to its Maclaurin series. 89. 3 + + (b) Graph the function in part (a) and comment on its 2! 3! 41 behavior near the origin. 90 + .. 97. Use the following steps to prove Theorem 17. 1 . 2 3 . 2 '5 . 25 7 . 27 (a) Let g(x) = En=o(")x". Differentiate this series to show that 91. Show that if p is an nth-degree polynomial, then kg(x) g'(x) = -1 P" (x) izo i! (b) Let h(x) = (1 + x) *g(x) and show that h'(x) = 0. (c) Deduce that g(x) = (1 + x). 92. Use the Maclaurin series for f(x) = x/(1 + x') to find fuD) (0). 98. In Exercise 10.2.62 it was shown that the length of the ellipse x = a sin0, y = b cos0, where a > b > 0, is 93. Use the Maclaurin series for f(x) = x sin(x) to find f(203)(0). 94. If f(x) = (1 + x))", what is f(58)(0)? L = 4a (#/2 V1 - e sin20 de 95. Prove Taylor's Inequality for n = 2, that is, prove that if If"(x) | M for | x - a| Ed, then where e = va2 - b2/a is the eccentricity of the ellipse. Expand the integrand as a binomial series and use the | R2 (x) | |x - ap for|x - alEd result of Exercise 7. 1.56 to express L as a series in powers of the eccentricity up to the term in e