Math 150B Homework 1 Answers

100 $ due x (log,(x3) (e) do tan (1) (I T/A -1 = 2x 10g, x3 + l2 2xx3 = = 1 4 1/2 x 1) x2+1 xx 2x log In 2 (f) 105'(5") - ln5.5" (b) Tx of log, x+1 = 1-52x = Ix log los, (x+1) 2.(a) dxf x = d Sxlnx e dx da dr ln 3 ln3 d ln x2 - ln(x+1) =(21/L lnx + F) 1 ( X 2x (b) of (sin x)* d exin(sing) e en 3 dx = ln3 1/3 ( 2/2 = [en (sin x) x cos x sin (c) d & (3) - [ n +x cot x " (2x+1) = All Jx = is ((norx-3) =[3ln(2x+1) + 2x+ 6x ](2x+1)" x63 " - (xln(3)-3) 3.(a) dx u x x ' - 1 du = 3x2 dx . 155" du (d) d sec'(x') day. = x 2 de dx 11.54 3 ln 5 2x = 5x3-1 1. +C 3 In 5 2 x/x-1

(b) dx u=ln (f) 3 dx =1-3 = dx du= dx /1-92 I -(3*)2 = S ln x dx x ln 2 In 3 / 1 - 12 u=32 1/2 Su du du = In 3.3 dx = 1. sin'( 3*)+c ln3 (lnx)2 + du = 3 dx = ln3 2 ln 2 (c) up dx u=x2 4(a) lim as+a x xx = lim et lnx = lnx V1-x4 du x dx - 2 e L HOP 1-(x) = e .e 1 = (b) lim (1-1-" lim 3xln(1-1) x-s - 00 x e 11-42 x-s+oa ln(1-k) sin'tx2)+c 2 = = e L'HOP 1/22 sin x extra 1-kx 3x4 = (d) S dx u=ws -1 x-1 1 + cos' x du - - sin x dx 3x' I -3x -3x = x-1 x(1-1') e -3 1-1/2 = e-3 tant ( cos x ) + c (e) Sxp dx (c) lim (2+7))* = e e ln(2+7") x 818 x/9x2-1 L'HOP G7.7 = 7.8m7 = e limotoo = 2+7' *(2,+1) x/(3x):- e h 7 sec+ (13x1) +

lim Infx (d) lim (5x) bnx = ex-o+ ln lim 1/2 lax x-so+ . e x->++ bast 2e ee 5 (a) lim 2x = 2-500 08 x3 a L HOP L'HOP , Hep In2.2 lim (l22)'.2 ( (n2)' 2 " = = = 8 x-> 8 3x2 6x 6 f (x) >> g(x) (b) lim ln x 3 8 = 2500 (lnx)2 8 L'HOP 3x2 lim 3 = = 0 2 lnx 2 enx x g (x) >> f (x) 6.00 S x2 COS 2x dx u x x 2 dv. = Scos 2x dx du = 2x dx V= sin 2x 2 = uv-S V du = x2 sin 2x - S sin 2 2x 2x dx U = x dv = P sin 2x dx 2 du = dx V = - cos 2x = x2 sin 2x + x COS 2x - s cos 2x dx 2 2 2 2 x2 sin 2x x cos 2x sin 2x + - = + 2 2 4

(6.(b) u=sin'(x) du-fide 2 a = x sin'(x) - Special 1-x2 dx a = 1 - x 2 du = -2x dx sim'(x) + 1/2 S 1/1 du = x =x u=lnx x' du= dx = - lnx = x (d) u.clnx) du = 2 ln x dx V = x = x(lnx) S 2 lnx dx u = ln x du dx V = x x(ln x)2- - 2xlnx+2x+c

(e) S ex -x cos 3x dx = S u=e" dv = S w s 3x dx du . . e x dx sin 3x V = 3 = e-x 3in 3x I e-x -x + sin 3rd dV = fsin 3x dx 3 3 dU .-e- dx V = - cos 3x -x -x 3 e sin 3x 1/m -e-" x = + [ cos 3x 3 - 1s 3 e**cos 3x J 3 -x S ex -x sin 3x e cos 3x = - - 3 9 1/4 S 10 S e-x = sin 3x ex cos 3x - 9 3 9 S = 3e" sin 3x e cos 3x c 10 10 (f) S mix dx u = 5x du = = s sin (w w du 2/x dx = 2/1/2 2u dx w du = dx = 2u cos u + S20034 du U= zu dv sin u du = - 2u cos u du 2 du V cos . u = - 2 5x us Sx

= x u=sin dx du = cos = x dx SC1-U2)2 S1+04-242 du = (b) S in 3 x dx = S sin x sin'x cos'x dx u = cos x cos'x -du=sin x dx = dx cos2 du. (u+c - = + cos x cc) S sin** c os'x dx = I 1 - cos 2x , 2 * 1 + cos 2x dx 2 = 19, cos22x dx = 1 S sin 2x dx = 1/4 S 1 - 00 4 X dx 2 = cos 4 x dx = 1/2 8 - sin 32 4x +

dx = Speck tan x tan'x dx u = see : = (sec2'x-1) dx du - sec x tan x dx = Sub-1 du sec x + c 3 = ST dx x =2 sin Q dx = 2cos o = x = s s n o 2 sin 0 = = x ,0 O = sin ( s) /4-x2 = do 12 = = 120 + sin 20 + C = O+C 2 14-x2+c + = 2

9.60 $ x x + dx = = x dx 2 P x 2 + 9 dx x - 9x x(x2-9) x (x-3)x+3) = = S A B c + + x x-3 x+3 + dx x x-3 x+3 = - ln / x 1 + ln/x-31. ln/x+31 + x 2 9 = A(x-3)(x+3) (x)(x+3)+ C (x)(x-3) When x=0; 9 = -9A A=-1 When x 3 ; 18 = 18B = B=1 When x = 3 ; 18 = 18C =) C =

(b) x+3 dx = State x-1 1. dx (x+1)(cx-1) x+1 = ln/x-11 +3 ln/x+1/ + 1 + x+1 x+3 = A + B + (c) (x+1)(x-1) x-1 x+1 (x+1)2 x + 3 = A(x+1)2 + B(x+1)(x-1) + (x-1) When x 1; 4 =4A =) A=1 When x = -1; 2==2 =) C = -1 When x = O; 3A-B-C - 3 =1-B+1 => B = 1