1. Evaluate the following integrals. ( a) VI - x - dx. (S.R: u = V1 -x.) (b) / (Inx)'dx. (IBP) Answer: x((Inx)3 - 3(Inx)2 + 6Inx -6) + C (c) xledx. (S.R + IBP) Answer: (20-De + C (d) tan'x secada. (Trig Integral) Answer: Sec'z - sec x + C' ( e ) (ex - 2) (e2x + 1) dx. (S.R + PF) Answer: } In |ex-2|-10 In(e27 +1)-? tan-1 ex +C (f ) 1 dx. (Trig Substitution) Answer: In(Vx2 - 4 + x) + C Vac2 - 4 x + 2 (g) ac (32 - 4x + 3) dx. (PF) (h) 2 2 - 1 dx. (S.R - GPR) 2. Evaluate the following integrals. ( 2 ) L x2 sin x 1+ x6 do. Answer: 0, since it is an integral of an odd function on (- 7, 2). ( b ) 10 In x dx. Answer: oo, divergent 3. Find the area of the region bounded by the curve x = y2 - 4y and x = 2y - y. Answer: 9 4. Use shell method to find the volume of the solid obtained by rotating the region bounded by y = x2, y =0 and x = 1 about the x-axis. Answer: 5. Use disk method to find the volume of a sphere of radius R. Answer: 2 T R36. Find the surface area obtained by rotating the curve y = sin m: (0 g a: g 1) about the m-axis. Answer: 2V1 + 7T2 + %1H(71' + V1+ 7T2) 2 7. Find the average value of f(z) = cos4zr sing: on [0,?1'). Answer: 3' 8. Determine whether the following sequences are convergent or divergent. If the sequence is convergent, determine its limit. 6\" 6\" 1 82\" (a) (Ln = m. ADSWBI'I (In = W ) 1 _ (1)\"(n3 + 4712 + 5) (b) 5\" n6+7n4+n2+1 Answer: 0, since |bn| ) 0. Answer: divergent, since an_1 > 4 and C2\" > 4. 9. Determine whether the series is convergent or divergent. If it is convergent, specify whether it is absolutely convergent or conditionally convergent. (a) Z n . (Test for Divergence) Answer: divergent n2 1 n=2 5 (b) E m. (Comparison Test) Answer: convergent (c) 2 n + 5 . (Ratio/Root Test) Answer: convergent n:1 5n 0 3n ((31) Z: (20\". (Root/Ratio/ Comparison Test) Answer: convergent n21 DO 0 4 e sm( n) . Comparison Test: on S in Answer: Absolutel conver ent 4n 4 y g 00 _1 n 2 (f) E %. (Alternating Series and Comparison Tests) n Answer: conditionally convergent DO 1 n (g) 1117123. (Ratio / Root / Comparison Test) Answer: convergent n: 1 1 M8 (11) nlnn' (Integral Test) Answer: divergent i\"? [\\3 10. Use definition to find the MacLaurin series of the function f(x) = In(1 + 5x), and Answer: (-1) n-15n determine the radius of convergence. M an, R = =. n n=1 11. For each of the following power series, find the radius of convergence and the interval of convergence. (a) n! Answer: R = 00, I = (-00, 00) n= 0 (b) > "an". Answer: R = 0, I = [0, 0] n=1 (c) (-1)n Vn + 3 6n -(2 + 5)". Answer: R = 6, I = (-11, 1) n=1 12. Let f(x) = tan-x. (a) Find a power series representation for f'(x). (b) Use the power series in (a) to find a power series representation for f(x). Answer: see Example 2(3) in the lecture outline of Section 11.9. 13. Evaluate / cos(x?) dx and cos(x2) dx as infinite series. (Use the MacLaurin series of cos(x) and then the FTC - Part 2.) 14. Find the sum for each of the following series. (a) E(-1)n (7 / 6) 2n + 1 (2n + 1)! . (Use the MacLaurin series of sin x) Answer: 2 n=0 (b) > (-1)n (Use the MacLaurin series of ex) Answer: e-1 - 1 n=1 n! 00 23n+1 (c) 32n+1 (Geometric series) Answer: 6 n=0 3 (d) y n (n + 3) (Telescoping series) Answer: 11 n=115. Find a power series representation for each of the following functions. Also find the radius of convergence and the interval of convergence. (a) f(2) = 1 + 2x2 . 1 OO Answer: f(x) = x 1 - (-2x2 ) C(-2) ninth, R = $21 (- 72: V2) n= 0 (b) f(20) = - 1+x 1 - x 2 Answer: f(a) = -1+ = 1+ ) 2x7, R=1, (-1, 1) 1 - x n=1 d pac2 + cos x sin t 16. Evaluate dt. (FTC - Part 1) da J5 t 2 + 3 sin(x2 + cos x) Answer: (x2 + cosx)2+3 (X - sin x) sin x 17. Determine whether or not the improper integral dx is convergent. Vx sin x Answer: convergent, since 0 and dx = 2VT is convergent. Vx Vx