solutions ws 7.5 pfd calc ab

Section 7.5 Partial Fractions 137 E& Section 7.5 Partial Fractions i ; | } ! 5 5 A B 4243 A B c a L = =24 A - g IXZ“lOX xx—10) x x-10 2(x—5)3 ):45+(x~5)3+(x~5)3 3 - ek 4, = = = = 3 410 x(x2+10) x x2+10 xP+4x+3 (x+Dx+3) x+1 x+3 1 t F 2x—=3 2x -3 A Bx+C =2 x—2 A B I I { i I s l6x __lex A4 B C g Bl _A Bx+C DitE T -1 x(x—10) x x2 x-10 TxxTH1)?2 ox o xTH1 0 (212 i R S 1 __A . B P 1 __A . B 0 & Txl-1 (A D -1 x+1 o x—1 "4x2-9 (x-3)(2x+3) -3 2x+3 1=Alx—1) + Bx + 1) 1= AQx + 3) + B(2x — 3) y Whenx = —1,1 = —24,4 = -}, Whenx =3,1 =6A,A =1 i Whenx = 1,1 =2B,B =}, Whenx = —3,1 = —6B,B = —+. 1 1 1 1 1 1 1 1 1 i - = —= e = - _ ‘ thldx 2fx+1d‘ 2fx~1d‘ [4x2—9d" 6[ FrariL f2x+3d”] : 1 1 I8 = —gmp 1]+ il 1]+ C = J5linl2x = 3~ nfar + 3] + € i B iR e 1o x—1 1. |2x—3 . == s 2 | L z‘“x+1| ¢ 121"2x+3’+c 3 3 A B x4l (1 9’x3+x‘2"(x—1)(x+2)‘xvl+x+z lo'fxz+4x+3dx_fx to e o2 3 o1 3=(x+2)+Bx-1) Whenx =1,3=34,A=1. Whenx = -2,3 = -3B,B= —1. 3 1 1 J’x2+x—2d17_]’x—1dx_fx+2dx =lnjx~ 1| -Inx+2|+C = 1 —lnx+2‘+C 1. ’5*): _ 5 —x __A 3 B P+ x—-1 (2x—-Dx+1) 2x—1 x+1 S5—x=Alx+1)+B2x-1) Whenx = 3,5 =34,4 = 3. Whenx= —1,6 = —38, B = —2. Sk 1 1 ‘[2x7+x~1dx‘3f2x—ldx—zfx-fldx =%ln|2x* 1| =2lx+1}+C

138 Chapter 7 Integration Techniques, L’'Hébpital’s Rule, and Improper Integrals 2 g 3x Tx—2 A B+C 12'x(x~1)(x+1):; x—1 x+1 3x2=Tx —2=Ax*~ 1) + Bx(x + 1) + Cx{x — 1) Whenx=0,-2= —A,A =2 Whenx=1,-6 =2B,B= —3 Whenx = —1,8 =2C,C = 4. 27, - f3" Eo i ndd PN J dx—3f—dx+4f i—x =2Injx| = 3In|x— 1| +4Injx + 1| + C xP+12x+12 A B G . =44 + B x(x+2)x—2) x x+2 x-2 X2+ 12x + 12 = A(x + 2)(x — 2) + Bx(x — 2) + Cx(x + 2) Whenx = 0,12 = —4A,A = —3. Whenx = —2, -8 = 8B,B= —1.Whenx = 2,40 = 8C,C = 5. X4 12+ 12 1 1 1 f e dx-sfx_zdx—fx+2¢x—3fxdx =5Inlx — 2| = Injx + 2| = 3In|x| + C B —x+3 2+ 1 A B =T T I =x—1+ + W N e = Wrea Tt 2+ 1=Ax — 1) + B(x + 2) Whenx= ~2, 3= BA,A =1 Whenx = 1B =385~ 1. 5 2 B | =%—x+ln|x+2|+ln]x—ll+C=%—x+ln|xz+x~2|+C i 2 — 4 - 15x + 5 X+ A B | 2 = 2x + =2x + , = =3 Zremar+) Ftioatiez x+5=A(x+2) + Bx—4) Whenx = 4,9 = 64,A = 2. Whenx = —2,3 = —6B, B = —1. 2x3—4x—15x+5dx 2_1/2 x2—2x—8 x=4 x+2 =xz+%1n‘x74t 7%ln|x+2| € 5 2t2, . A B AP t2-1 A B C “x(x—4) x-4 x ToxAx+ 1) x xz x+1 x+2=Ax+B(x—4) 4x2 + 2x — 1 = Ax(x + 1) + B(x + 1) + Cx? When x = 4,6 = 44,A = 3. Whenx =0,B=—1.Whenx= ~1,C = 1. Whenx = 0,2 = —4B,B = —3. Whenx = 1,4 = 3. (4x2 + 20— 1 31 1 xv2 o ([32_172), f P dx—f[;‘;*,(+1]dx x* — 4x x—4 % 1 =%ln]x—4|—%1n|x|+c i 3ln|x|+x+ln]x+1]+c l+ lajx? + %*| + € x

140 Chapter 7 Integration Techniques, L'Hépital’s Rule, and Improper Integrals 53 x __A B Cx+D T - D@+ D@xZ+ 1) 22— 1 2+ 1 a2+l x=AQx + 1)(4x? + 1) + B2x — 1)(4x? + 1) + (Cx + D)(2x — 1)2x + 1) Whenx = 3,3 = 44. Whenx = ~4, —1 = —4B Whenx = 0,0 =4 — B — D,and whenx = 1, 1 = 15A + SB + 3C + 3D. Solving these equations we have A = %,B = 3, C= —l ,D =0 fléx =14 U’Zx—ldx j2x+l J4x2x+1d"] :16 %I‘”Jrc x2—4x+7 _ A Bx+ C 2, (+ D2 —2x+3) x+1 x*-2c+3 X4+ T7=A"-2x+3)+ Bx + O)x + 1) Whenx = —1,12 = 6A. Whenx = 0,7 = 34 + C. Whenx = 1,4 = 24 + 2B + 2C. Solving these equations we have A=2,B=-1,C=1. 22 —dxt 7 1 =%+ 1 Jx’—x1+x+3dx‘2fx+ldx+J-x172x+3dx =2lnjx + 1] —%1n|x2—— 2x+3| + C %2 45 _ A Bx+.¢ x+1DE2—2x+3) x+1 x2—2x+3 A5 =AR2 -2 +3) + Bx+ O + 1) 25. =(A+Bx2+(-2A+ B+ COx+ (3A + ) Whenx = —1,A = 1. By equating coefficients of like terms, we have A + B =1, =24 + B+ C=0,3A + C = 5. Solving these equations we have A = 1,B = 0,C = 2. x5 1 1 J:fi—fl+)c+3dx_J’x+ld)(-i—zf(x—1)24r2d)c =lnjx+ 1| + flarctan(x‘;il) +C x*+x+3 Ax+B Cx+D 3 A B . = + 4 = T E s A P T A Ne D malties X2+ x+3=Ax+B)(2+3)+Cx+ D 3=A(x+2) +B2x+ 1) =Ax* + Bx2 + (3A + O)x + (3B + D) Whenx = —5,A =2 Whenx = —2,B = — By equating coefficients of like terms, we have A = 0, dr = w- [y B =1,34A + C=1,3B + D = 3. Solving these equations o 2x2 + 5x + 27 b 2x + 1 0x+2x wehaveA=0,B=1,C=1,D=0. 1 s 1 =[ln|2x—1!*ln|x+2|:| X X dx:j[ X :| 0 Frero P R =2 = Larctz\ni~—1—+C V3 V32w +3)

p Section 7.5 Partial Fractions 141 x—1 A B c e = S 2 xAx+ 1) x x1+x+l ) x—1=Ax(x + 1) + B(x + 1) + Cx? f' Whenx =0,B= —1.Whenx=—1,C= -2 Whenx=1,0=24 + 2B + C. Solving these equations we have A = 2, B=-1,C=-2 P x-1 f,xl(x%—l) f,xdx J: dx—Z s = [21n|x| gl 2Infx + 1|] X “ 1 5 +1] X =2 x x4 1 5 4 —21n§—5 x+ 1 *é+Bx+C Txx24+ 1) x x2+1 x+1=AK*+ 1)+ (Bx + O)x Whenx = 0,A =1.Whenx=1,2=24A+ B+ C. Whenx=—-1,0=24 + B— C. Solving these equations we have A=1,B=-1,C=1. 2 2 Xt 1 1 fx(xz+1)dx f Pl fxz ldx+J;x2+ldx 2 = [lnlx| = %ln(xZ +1) + arctanx] 1 1. 8 = —51n5~4+arctan2 = 0.557 x?—x 2x + 1 3xdx 9 = L = -3 - ——+ SO'J:)xz+x+ldx jdx fxz+x+l 2 fxl—6x+9 3 lns =3 x—3 € ! 4,0): 3In|4 - 3| - :[x—ln]xz+x+ 1]:|0 “9 ol | =1-I3 J -6 10 x—1] 1 2 7 +-+ - + x | Tx x—1 2-12 € 7 1 -E C=1=C=2 31n2

Section 7.5 Partial Fractions 3 cosx 1 1 A B 3 = i =24+ = 4= . du = sec? 4 jsin2x+sinx-2fl 3fu1+u—2d“ 42, WD) et ur u =tanx, du = sec’ xdx - = A+ 1) + —n +;+C 1=A(u+ 1)+ Bu Whenu = 0,A = 1. | —1 + sinx Whenu = —-1,1= -B=>B = —1. n " 2 + sinx J» et v de =J’ 1 W (From Exercise 9 with u = sin x, du = cos x dx) tan x(tan x + 1) ulu + 1) 1 1 _J’(u u+l)du =Infu| —Infu + 1| + C _ u —lnu+1 +C tan x - ’tanxi—l 4 43. Letu = e, du = e*dx. 1 A B G-Du+d u-1'u+a 1=A(+4) + Bu— 1) Whenu = 1,A = % Whenu = —4,B = ~ u = &5, du = e*dx. e* 1 I(e"—l)(e‘+4)dx:f(u—l)(u+4)du 1 1 1 -flmdu-fmd“) 1 ju—1 _gln1u+4’+c 1 fer—1 75'"je*+4‘+c 4. Letu = e, du = e*dx. 1 A Bu+ C (ul+1)(u—l):u—l W+ 1 1 =AW + 1)+ (Bu+ O~ 1) Whenu = 1,A = % Whenu=0,1=A—-C. Whenu = —1,1 = 24 + 2B — 2C. Solving these equations we have A = 4, B = —-%, C= —']i, u = e, du = e*dx. er 1 f(ez:+1)(ex- 1)“=J(u2+ D= D 1 1 u+1 =E(fiu—1d”_fu2+1d“> l( 1 ) = — G ez - + 7 Inju — 1| 2ln|u + 1| —arctanu | + C =%(21n|e"— 1| = Infe? + 1] — 2 arctan e?) + C

144 Chapter 7 Integration Techniques, L'Hépital’s Rule, and Improper Integrals 1 A B 45'x(a+bx)_;+a+bx 1 = Ala + bx) + Bx Whenx = 0,1 =aA=A = 1/a. Whenx = —a/b,1 = —(a/b)B =B = —b/a. i U1 b fx(a+bx)dx7aJ’<x a+bx)dx = %(ln{x[ ~Inja + bx|) + C = gl ‘+c a |a+ bx % A B . = + 4 (a+bx)> a+bx (a+bx)? x=Ala + bx) + B Whenx = —a/b,B = —a/b. Whenx =0,0=aA + B = A = 1/b. x _ 1/b *a/b) J(a+bx)zfl7j(a+bx+(a+bx)2 ax 1 1 a 1 -;J‘aflzxdx_zj(awvc)?dx =L af 1 = Inja + bx| + bz(a = bx) $4€ 1 a = fi(a_rb; + Inla + bxf) +C 3 49.A:f 1 dx =3 i x(x+ 1) Matches (c) ¥ ax \ e 51. (a) V= WJ:) (;-—2 T l) dx = 47 (]—“(xz = ])zdx iR 1 = 411—J:] (-‘—-xl 1 —-—(X2 e l)z)dx 1 = 477[arctan X §<arctan x+ x 3 = ZW[arclanx - 2—] = 277|: x*+ 1o —CONTINUED— x Pl | a+x 1= =Ala + x) + Bla — x) =a,A=1/2a. Whenx = —a,B = 1/2a. 1 1 1 1 == + faz—xzdx 2a (a~x a+x)dx = -l—(*ln|a — x| +lnja+x]) + C 2a 1. la+x =1 + 2a —x € 1 A B C 48'x2(a+bx)7x+x2 a + bx 1 = Ax(a + bx) + Bla + bx) + Cx? Whenx =0,1 =Ba = B = l/a. Whenx = —a/b, | = C(a?/b*) = C = b*/a’. Whenx=1,1=(@+bA +(a+bB+C = A= —b/a? 1 _ —b/a* 1/a J’xz(a+bx)d’(7f( x +x2+ b 1 b == — —+ Znla + bx| + pa In|x] Pl Inja + bx| + C b/a? )dx a + bx 1 + ———+%Ina bx+C ax a X 1 b x T e & a+b‘+c 14 |4+ x|} = [Zx - — ln‘4 = j ]0 (From Exercise 46) i :67%1117@*2.595 i -3 -2 -1 1 2 3 (partial fractions) 3 )] (trigonometric substitution) o 10 arctan 3 — i] = 5963