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=\"F*$\"3h=z=#yjn@\"F*7$$\"3Od()H5%foD\"F*$\"3/CS8j$[?H\"F*7$$\"30wj\" *>rz>8F*$\"3KjV)Gix3O\"F*7$$\"3-0.)oriTQ\"F*$\"3y?0Z(Rx>V\"F*7$$\"37!= ]5(3i\\9F*$\"3uHb@F(z]]\"F*7$$\"3y/p[_'*y4:F*$\"3'f\\wF.SId\"F*7$$\"3! )))p3Dw#Rd\"F*$\"3upSjJgNY;F*7$$\"3Jv/FvtdP;F*$\"31$*=jgj2?F*7$$\"3aUI6Wji(*=F*$\"3[NN.?Q1L?F*7$$\"3O] vK*eC&f>F*$\"3*ysQi7u36#F*7$$\"39W&Gz3&3E?F*$\"3k%*y,t+C'>#F*7$$\"3_!y mu$[\"H4#F*$\"37&3q'\\BF*$\"3Rf'>oBDIk#F*7$$\"3\"*\\j/Ams3CF*$\"3j*[Du<7?t#F*7$$\"3g:4 k2PhwCF*$\"3uf+26e\"z$GF*7$$\"3*4pTq8J/a#F*$\"3oEsG1OhTHF*7$$\"3O>.nf@ j.EF*$\"3=#e#\\pK#*[IF*7$$\"3WKY=fM)om#F*$\"3[Tci*RP<;$F*7$$\"3@:'H4y& RJFF*$\"37*=W')G4NG$F*7$$\"33B/Bhxx*z#F*$\"35C#yJVP`!*H F*$\"39[tnS&Gr)QF*7$$\"3qCRD`+bcIF*$\"3'=XMB\"yU*4%F*7$$\"33#e[Y>0e6$F *$\"3`:Pb&)\\JWVF*7$$\"33++me))4$=$F*$\"3Q8\"ROwd)**\\F*-Fjz6&F\\[lF_[ lF][lF][l-F$6$7S7$F(F(7$F.F.7$F3F37$F8F87$F=F=7$FBFB7$FGFG7$FLFL7$FQFQ 7$FVFV7$FenFen7$FjnFjn7$F_oF_o7$FdoFdo7$FioFio7$F^pF^p7$FcpFcp7$FhpFhp 7$F]qF]q7$FbqFbq7$FgqFgq7$F]rF]r7$FbrFbr7$FgrFgr7$F\\sF\\s7$FbsFbs7$Fg sFgs7$F\\tF\\t7$FatFat7$FftFft7$F[uF[u7$F`uF`u7$FeuFeu7$FjuFju7$F_vF_v 7$FdvFdv7$FivFiv7$F^wF^w7$FcwFcw7$FhwFhw7$F]xF]x7$FbxFbx7$FgxFgx7$F\\y F\\y7$FayFay7$FfyFfy7$F[zF[z7$F`zF`z7$FezFez-Fjz6&F\\[lF][lF_[lF][l-%% TEXTG6&7$$\"\"&!\"\"$\"#F!\"#Q*y~=~sin~x6\"Fiz-%%FONTG6$%*HELVETICAG\" \"*-Fc^n6&7$$\"\"#Fh^n$\"#bF[_nQ-y~=~arcsin~xF]_nFjjmF^_n-Fc^n6&7$$\" \"'Fh^n$!\"$F[_nQ\"xF]_n-Fjz6&F\\[lF^[lF^[lF^[lF^_n-Fc^n6&7$F``nF^`nQ \"yF]_nFc`nF^_n-%*AXESTICKSG6$7'/$!\"&Fh^n%%-p/2G/$!&J=$F_an%#-1G/F^[l %\"0G/$\"&J=$F_an%\"1G/Ff^n%$p/2GF\\an-F__n6$%'SYMBOLGFb_n-%+AXESLABEL SG6%%!GFcbn-F__n6#%(DEFAULTG-%%VIEWG6$;F^anF^`nFjbn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 257 " " 0 "" {TEXT -1 17 "Since the graph " }{XPPEDIT 18 0 "y = sin*x;" "6# /%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 14 " has the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 40 " as a tangent at the origin, \+ the graph " }{XPPEDIT 18 0 "y = arcsin*x;" "6#/%\"yG*&%'arcsinG\"\"\" %\"xGF'" }{TEXT -1 20 ", also has the line " }{XPPEDIT 18 0 "y=x" "6#/ %\"yG%\"xG" }{TEXT -1 30 " as a tangent at the origin. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " } {XPPEDIT 18 0 "y = arcsin*x;" "6#/%\"yG*&%'arcsinG\"\"\"%\"xGF'" } {TEXT -1 56 " on its own suggests the shape of \"half an hour-glass\". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 298 369 369 {PLOTDATA 2 "6*-%'CURVESG6$7`q7$$!3%****za&))4$=$!#=$!3iCPG9))z**\\F*7 $$!3P2\"f)e)=e6$F*$!3\"*G)[[z\"QWVF*7$$!3QcMa$*)ys0$F*$!3N+Pn/B/-TF*7$ $!3sg@/!eW9*HF*$!3)=WiI#ez*)QF*7$$!3_lj(R(Qk?vMFF*$!3G$*Hx;n0!H$F*7$$!39)f*3u)p#pEF*$!3#fx@NC8h;$F*7$$!3[22?'o(* Rg#F*$!3!HY>kue&\\IF*7$$!35^_)z@do`#F*$!3=xG,O$*oNHF*7$$!3Oed.d)>xZ#F* $!3F)HcdBx'RGF*7$$!3RvCD&>X6T#F*$!3byG-&H9dt#F*7$$!3w[:^rrHWBF*$!3D+/y $=Z^j#F*7$$!3E\"e**Rrw)zAF*$!3@ayi`TSTDF*7$$!3)3f))pww8A#F*$!33U&e(\\% p'eCF*7$$!3g2lvrY\"=:#F*$!3#y\"zS`[$HO#F*7$$!3!)z)>EX')G4#F*$!3ea6l[*e QG#F*7$$!3-VM\"[#*QV-#F*$!3aBj'RrwR>#F*7$$!3[ZJxZFmj>F*$!3p<=[4'Gh6#F* 7$$!3c7K!\\O\"4(*=F*$!3l&fpl`(RK?F*7$$!3#=M]cm*pL=F*$!3I\"Rsm(*eT&>F*7 $$!3o=$4S^\"F*$!3gfGt=^$yd\"F*7$$!3Sec#efG+X\"F*$!3X_gD7fE>8F*$!3!3G)=(*QHg8F*7$$!3[ynE T)*pc7F*$!3_z@V$ou=H\"F*7$$!3E/Ds')3B(=\"F*$!3W'**>t`Qm@\"F*7$$!3-,zIT *4[7\"F*$!3K_2`F9k\\6F*7$$!3m'y$feA;e5F*$!3C)G0<;!py5F*7$$!3/_WhAbpx** !#>$!3>\"fet#o([,\"F*7$$!3_g2OCPW<$*F_v$!3%RxM#G$4fX*F_v7$$!3Ox)HoB)>' p)F_v$!3+c$*Q_\"z\"3))F_v7$$!3e)\\4')pT(F_v$!3?8O15\")f![(F_v7$$!3A4Lw[/EZnF_v$!3W*HM!R\"Q))z'F_v7$$! 3%\\Zn!H^52hF_v$!3s!owA@2_9'F_v7$$!3)p*p*\\&QX_aF_v$!3'RrmH9w%zaF_v7$$ !3m<@+CMA.[F_v$!3DJ>1-6k@[F_v7$$!36.?Q%3[m?%F_v$!3\"Qd*[%G!**=UF_v7$$! 3D8[o@a!H_$F_v$!3Tc:O&QP,`$F_v7$$!3!3*H@LyN6HF_v$!3`:([AhAF_v7$$!3q#QYfnP_j\"F_v$!3G94=+y&fj\"F_v7$$!3N :PTf%QFU*!#?$!3SE$er>:TU*Ffz7$$!3WvVRn3=AMFfz$!3%po@pzYAU$Ffz7$$\"3w'H <9n0&)Q$Ffz$\"3A%*[nt'p&)Q$Ffz7$$\"3i]k?$32df*Ffz$\"3u%zg(e5;(f*Ffz7$$ \"3m*p*p#*4aQ;F_v$\"3[kgP$\\l#R;F_v7$$\"3;n9N@vPCAF_v$\"3I?4+4>>EAF_v7 $$\"3cxc3Wpd\"*GF_v$\"31KE]f(ob*GF_v7$$\"3<-::['eg`$F_v$\"3CO\"QMAsLa$ F_v7$$\"3CI8\"zK>,=%F_v$\"3uY*e6GGA>%F_v7$$\"3j15jC1\"=#[F_v$\"3=h0\"= XW/%[F_v7$$\"3\"4cc>2v#QaF_v$\"3Yyi[=_3laF_v7$$\"3_&eY`Y*o/hF_v$\"3?^= ,SeuUhF_v7$$\"3$=t3S8!)F_v$\"3Bb$R,ia05)F_v7$$\"3sfCiC\\;#o)F_v$ \"3gC249=f$z)F_v7$$\"3m*f:9SCFK*F_v$\"3gdT)Q/K9Y*F_v7$$\"3?;S^QZsh**F_ v$\"3/>O%R8&>85F*7$$\"3[3'GnI;H1\"F*$\"3)H(4x)HKP3\"F*7$$\"3XM(yL\")*Q C6F*$\"3%))y8sL#>\\6F*7$$\"37Y`%H4Zt=\"F*$\"3h=z=#yjn@\"F*7$$\"3Od()H5 %foD\"F*$\"3/CS8j$[?H\"F*7$$\"30wj\"*>rz>8F*$\"3KjV)Gix3O\"F*7$$\"3-0. )oriTQ\"F*$\"3y?0Z(Rx>V\"F*7$$\"37!=]5(3i\\9F*$\"3uHb@F(z]]\"F*7$$\"3y /p[_'*y4:F*$\"3'f\\wF.SId\"F*7$$\"3!)))p3Dw#Rd\"F*$\"3upSjJgNY;F*7$$\" 3Jv/FvtdP;F*$\"31$*=jgj2?F*7$$\" 3aUI6Wji(*=F*$\"3[NN.?Q1L?F*7$$\"3O]vK*eC&f>F*$\"3*ysQi7u36#F*7$$\"39W &Gz3&3E?F*$\"3k%*y,t+C'>#F*7$$\"3_!ymu$[\"H4#F*$\"37&3q'\\BF*$\"3Rf'>oBDIk#F*7$$\"3\"* \\j/Ams3CF*$\"3j*[Du<7?t#F*7$$\"3g:4k2PhwCF*$\"3uf+26e\"z$GF*7$$\"3*4p Tq8J/a#F*$\"3oEsG1OhTHF*7$$\"3O>.nf@j.EF*$\"3=#e#\\pK#*[IF*7$$\"3WKY=f M)om#F*$\"3[Tci*RP<;$F*7$$\"3@:'H4y&RJFF*$\"37*=W')G4NG$F*7$$\"33B/Bhx x*z#F*$\"35C#yJVP`!*HF*$\"39[tnS&Gr)QF*7$$\"3qCRD`+bcIF *$\"3'=XMB\"yU*4%F*7$$\"33#e[Y>0e6$F*$\"3`:Pb&)\\JWVF*7$$\"33++me))4$= $F*$\"3Q8\"ROwd)**\\F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`[mF_[m-% %TEXTG6&7$$\"\"#!\"\"$\"#[!\"#Q-y~=~arcsin~x6\"Fhjl-%%FONTG6$%*HELVETI CAG\"\"*-Fb[m6&7$$\"#PFj[m$!\"$Fj[mQ\"xF\\\\m-Fijl6&F[[mF`[mF`[mF`[mF] \\m-Fb[m6&7$$Fj[mFj[m$\"\"'Fg[mQ\"yF\\\\mFj\\mF]\\m-%*AXESTICKSG6$7%/$ !&J=$!\"&%#-1G/F`[m%\"0G/$\"&J=$Fj]m%\"1G7%/$Fj]mFg[m%%-p/2GF\\^m/$\" \"&Fg[m%$p/2G-F^\\m6$%'SYMBOLGFa\\m-%+AXESLABELSG6%%!GF`_m-F^\\m6#%(DE FAULTG-%%VIEWG6$;$!#LFj[mFe\\m;Fd^mF`]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 35 "The domain of the arcsine function " }{XPPEDIT 18 0 "arcsin*x;" "6#*&%'arcsinG\"\"\"%\" xGF%" }{TEXT -1 17 " is the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$ \"\"\"!\"\"F%" }{TEXT -1 32 ", and its range is the interval " } {XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F (F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "The arcsine funct ion has the following special values. " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "arcsin*0 = 0;" "6#/*&%'arcsinG\"\"\"\"\"!F&F'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(1/2) = Pi/6; " "6#/-%'arcsinG6#*&\"\"\"F(\"\"#!\"\"*&%#PiGF(\"\"'F*" }{TEXT -1 10 " , " }{XPPEDIT 18 0 "arcsin(-1/2) = -Pi/6;" "6#/-%'arcsinG6#,$* &\"\"\"F)\"\"#!\"\"F+,$*&%#PiGF)\"\"'F+F+" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(1/sqrt(2)) = Pi/4;" "6#/-%'arcsinG6#*&\"\"\"F(-%%sqrtG6#\"\"#!\"\"*&%#PiGF(\"\"%F-" } {TEXT -1 10 ", " }{XPPEDIT 18 0 "arcsin(-1/sqrt(2)) = -Pi/4;" "6#/-%'arcsinG6#,$*&\"\"\"F)-%%sqrtG6#\"\"#!\"\"F.,$*&%#PiGF)\"\"%F.F. " }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " arcsin(sqrt(3)/2)=Pi/3" "6#/-%'arcsinG6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#! \"\"*&%#PiGF,F+F." }{TEXT -1 11 ", a" }{XPPEDIT 18 0 "rcsin(-s qrt(3)/2)=-Pi/3" "6#/-%&rcsinG6#,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F/, $*&%#PiGF-F,F/F/" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin*1 = Pi/2;" "6#/*&%'arcsinG\"\"\"F&F&*&%#PiGF&\" \"#!\"\"" }{TEXT -1 11 ", a" }{XPPEDIT 18 0 "rcsin(-1) = -Pi/2 ;" "6#/-%&rcsinG6#,$\"\"\"!\"\",$*&%#PiGF(\"\"#F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The arc sine function is an " }{TEXT 259 12 "odd function" }{TEXT -1 38 ", tha t is, it satisfies the relation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(-x) = -arcsin*x;" "6#/-%'arcsinG6#,$%\"xG!\"\", $*&F%\"\"\"F(F,F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 287 12 "____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "deriv ative" }{TEXT -1 38 " of the arcsine function is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arcsin*x] = 1/sqrt(1-x^2);" "6 #/7#*&%'arcsinG\"\"\"%\"xGF'*&F'F'-%%sqrtG6#,&F'F'*$F(\"\"#!\"\"F0" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 14 "_____ _________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "This can be checked by implicit differentiation as follows." }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = \+ arcsin*x;" "6#/%\"yG*&%'arcsinG\"\"\"%\"xGF'" }{TEXT -1 12 ", we have " }{XPPEDIT 18 0 "sin*y = x;" "6#/*&%$sinG\"\"\"%\"yGF&%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "-Pi/2 <= y;" "6#1,$*&%#PiG\"\"\"\" \"#!\"\"F)%\"yG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"\" #!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {XPPEDIT 18 0 "cos*y;" "6#*&%$cosG\"\"\"%\"yGF%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 1;" "6#/*&%#dyG\"\"\"%#dxG!\"\"F&" }{TEXT -1 12 ", so that " }{XPPEDIT 18 0 "dy/dx = 1/(cos*y);" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&F&F&*&%$cosGF&%\"yGF&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "0 <= cos*y;" "6#1\"\"!*&%$cos G\"\"\"%\"yGF'" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "-Pi/2 <= y;" "6#1 ,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"yG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G *&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 16 " , the formula " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta=1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$s inGF'F(F)F)F)" }{TEXT -1 16 " implies that " }{XPPEDIT 18 0 "cos*y = sqrt(1-sin^2*y);" "6#/*&%$cosG\"\"\"%\"yGF&-%%sqrtG6#,&F&F&*&%$sinG\" \"#F'F&!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence \+ " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(1-sin^2*y);" "6#/*&%#dyG\"\"\"%#dxG! \"\"*&F&F&-%%sqrtG6#,&F&F&*&%$sinG\"\"#%\"yGF&F(F(" }{TEXT -1 13 ", t hat is, " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(1-x^2);" "6#/*&%#dyG\"\"\"%# dxG!\"\"*&F&F&-%%sqrtG6#,&F&F&*$%\"xG\"\"#F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 42 "The corresponding integration formula is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(1-x^2 ),x) = arcsin*x+c;" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&F(F(*$%\"xG\"\" #!\"\"F0F.,&*&%'arcsinGF(F.F(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 271 16 "________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "Maple \"knows\" these results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Di ff(arcsin(x),x);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%D iffG6$-%'arcsinG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\" \"\"F&*$,&F&F&*$)%\"xG\"\"#F&!\"\"#F&F,F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(1/(sqrt(1-x^2) ),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\" \"\"F'*$,&F'F'*$)%\"xG\"\"#F'!\"\"#F'F-F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%'arcsinG6#%\"xG\"\"\"%\"cGF*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "A definite integral involving arc sin" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "The following picture shows the graph of " }{XPPEDIT 18 0 "f(x) = arcsin*x;" "6#/-%\"fG6#%\"xG*&%'arcsinG\"\"\"F'F*" }{TEXT -1 30 " together with its derivative " }{XPPEDIT 18 0 "`f '`(x)=1/sqrt (1-x^2)" "6#/-%$f~'G6#%\"xG*&\"\"\"F)-%%sqrtG6#,&F)F)*$F'\"\"#!\"\"F0 " }{TEXT -1 3 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 57 "The arccosine funct ion has the following special values. " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "arccos*0 = Pi/2;" "6#/*&%'arccosG\"\"\"\"\"!F&*&%#PiGF&\"\"#!\" \"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(1/2) = Pi/3;" "6#/-%'arccosG6#*&\"\"\"F(\"\"#!\"\"*&%#PiGF( \"\"$F*" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "arccos(-1/2) = 2*Pi /3;" "6#/-%'arccosG6#,$*&\"\"\"F)\"\"#!\"\"F+*(F*F)%#PiGF)\"\"$F+" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arc cos(1/sqrt(2)) = Pi/4;" "6#/-%'arccosG6#*&\"\"\"F(-%%sqrtG6#\"\"#!\"\" *&%#PiGF(\"\"%F-" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "arccos(-1/ sqrt(2)) = 3*Pi/4;" "6#/-%'arccosG6#,$*&\"\"\"F)-%%sqrtG6#\"\"#!\"\"F. *(\"\"$F)%#PiGF)\"\"%F." }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(sqrt(3)/2) = Pi/6;" "6#/-%'arccosG6#*& -%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"*&%#PiGF,\"\"'F." }{TEXT -1 11 ", \+ " }{XPPEDIT 18 0 "arccos(-sqrt(3)/2) = 5*Pi/6;" "6#/-%'arccosG6#,$ *&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F/*(\"\"&F-%#PiGF-\"\"'F/" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccos*1 = 0 ;" "6#/*&%'arccosG\"\"\"F&F&\"\"!" }{TEXT -1 16 ", " } {XPPEDIT 18 0 "arccos(-1)=Pi" "6#/-%'arccosG6#,$\"\"\"!\"\"%#PiG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The following symmetry formula applies: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(-x) = Pi-arccos*x;" "6#/-%' arccosG6#,$%\"xG!\"\",&%#PiG\"\"\"*&F%F,F(F,F)" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 286 15 "_______________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "derivative" }{TEXT -1 40 " of the arc cosine function is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[arccos*x] = -1/sqrt(1-x^2);" "6#/7#*&%'arccosG\"\"\"% \"xGF',$*&F'F'-%%sqrtG6#,&F'F'*$F(\"\"#!\"\"F1F1" }{TEXT -1 1 "." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 16 "________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "This can be checked by implicit differentiation as follow s." }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = arccos* x;" "6#/%\"yG*&%'arccosG\"\"\"%\"xGF'" }{TEXT -1 11 " we have " } {XPPEDIT 18 0 "cos*y = x;" "6#/*&%$cosG\"\"\"%\"yGF&%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{XPPEDIT 18 0 "`` \+ <= Pi;" "6#1%!G%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Th en " }{XPPEDIT 18 0 "-sin*y;" "6#,$*&%$sinG\"\"\"%\"yGF&!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1;" "6#/*&%#dyG\"\"\"%#dxG!\"\" F&" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "dy/dx = -1/(sin*y);" "6 #/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&*&%$sinGF&%\"yGF&F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "0 <= sin*y;" "6#1\"\"!*&%$sinG\"\"\"%\"yGF'" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" } {XPPEDIT 18 0 "`` <= Pi;" "6#1%!G%#PiG" }{TEXT -1 16 " , the formula \+ " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta=1" "6#/,&*&%$cosG\"\"#%&thet aG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 16 " implies that " } {XPPEDIT 18 0 "sin*y = sqrt(1-cos^2*y);" "6#/*&%$sinG\"\"\"%\"yGF&-%%s qrtG6#,&F&F&*&%$cosG\"\"#F'F&!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "dy/dx = -1/sqrt(1-cos^2*y);" "6 #/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&-%%sqrtG6#,&F&F&*&%$cosG\"\"#%\"yGF&F (F(F(" }{TEXT -1 13 ", that is, " }{XPPEDIT 18 0 "dy/dx = -1/sqrt(1- x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&-%%sqrtG6#,&F&F&*$%\"xG\"\"# F(F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Maple \"knows\" this result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Diff(arccos( x),x);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%'a rccosG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$,&F%F %*$)%\"xG\"\"#F%!\"\"#F%F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding integration formula is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(1-x^2),x) = -arccos*x+c;" "6 #/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&F(F(*$%\"xG\"\"#!\"\"F0F.,&*&%'arccos GF(F.F(F0%\"cGF(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 272 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 2 ": " } }{PARA 15 "" 0 "" {TEXT -1 40 "In the previous section we stated that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(1-x^2 ),x) = arcsin*x+c;" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&F(F(*$%\"xG\"\" #!\"\"F0F.,&*&%'arcsinGF(F.F(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "This is not inconsis tent with the new result because " }{XPPEDIT 18 0 "-arccos*x;" "6#,$*& %'arccosG\"\"\"%\"xGF&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "arcsi n*x;" "6#*&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 1 " " }{TEXT 259 25 "only differ by a constant" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Indeed we have: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "arcsin*x+arccos*x = Pi/2;" "6#/,&*&%'arcsinG\"\"\"%\"xGF'F'*&%'arccos GF'F(F'F'*&%#PiGF'\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "for any number " }{TEXT 274 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 8 " and 1. \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 "Here are some examples: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin*0+arccos*0 = 0+Pi/2; " "6#/,&*&%'arcsinG\"\"\"\"\"!F'F'*&%'arccosGF'F(F'F',&F(F'*&%#PiGF'\" \"#!\"\"F'" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\" " }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " arcsin(1/2)+arccos(1/2) = Pi/6+Pi/3" "6#/,&-%'arcsinG6#*&\"\"\"F)\"\"# !\"\"F)-%'arccosG6#*&F)F)F*F+F),&*&%#PiGF)\"\"'F+F)*&F2F)\"\"$F+F)" } {XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(-1/2)+a rccos(-1/2) = -Pi/6+2*Pi/3;" "6#/,&-%'arcsinG6#,$*&\"\"\"F*\"\"#!\"\"F ,F*-%'arccosG6#,$*&F*F*F+F,F,F*,&*&%#PiGF*\"\"'F,F,*(F+F*F4F*\"\"$F,F* " }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(1/ sqrt(2))+arccos(1/sqrt(2)) = Pi/4+Pi/4" "6#/,&-%'arcsinG6#*&\"\"\"F)-% %sqrtG6#\"\"#!\"\"F)-%'arccosG6#*&F)F)-F+6#F-F.F),&*&%#PiGF)\"\"%F.F)* &F7F)F8F.F)" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\" " }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " arcsin(-1/sqrt(2))+arccos(-1/sqrt(2)) = -Pi/4+3*Pi/4;" "6#/,&-%'arcsin G6#,$*&\"\"\"F*-%%sqrtG6#\"\"#!\"\"F/F*-%'arccosG6#,$*&F*F*-F,6#F.F/F/ F*,&*&%#PiGF*\"\"%F/F/*(\"\"$F*F9F*F:F/F*" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(sqrt(3)/2)+arccos(sqrt(3)/2) = P i/3+Pi/6;" "6#/,&-%'arcsinG6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F--%'arc cosG6#*&-F*6#F,F-F.F/F-,&*&%#PiGF-F,F/F-*&F8F-\"\"'F/F-" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(-sqrt(3)/2)+ar ccos(-sqrt(3)/2) = -Pi/3+5*Pi/6;" "6#/,&-%'arcsinG6#,$*&-%%sqrtG6#\"\" $\"\"\"\"\"#!\"\"F0F.-%'arccosG6#,$*&-F+6#F-F.F/F0F0F.,&*&%#PiGF.F-F0F 0*(\"\"&F.F:F.\"\"'F0F." }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "'arcsin(0)+arccos(0)'=arcsi n(0)+arccos(0);\n'arcsin(1/2)+arccos(1/2)'=arcsin(1/2)+arccos(1/2);\n' arcsin(sqrt(2)/2)+arccos(sqrt(2)/2)'=arcsin(sqrt(2)/2)+arccos(sqrt(2)/ 2);\n'arcsin(sqrt(3)/2)+arccos(sqrt(3)/2)'=arcsin(sqrt(3)/2)+arccos(sq rt(3)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'arcsinG6#\"\"!\"\" \"-%'arccosGF'F),$*&\"\"#!\"\"%#PiGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'arcsinG6##\"\"\"\"\"#F)-%'arccosGF'F),$*&F*!\"\"%#PiGF)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'arcsinG6#,$*&#\"\"\"\"\"#F+- %%sqrtG6#F,F+F+F+-%'arccosGF'F+,$*&F,!\"\"%#PiGF+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&-%'arcsinG6#,$*&#\"\"\"\"\"#F+-%%sqrtG6#\"\"$F+F+F +-%'arccosGF'F+,$*&F,!\"\"%#PiGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }{XPPEDIT 18 0 "arcsin*x+arccos*x = Pi/2;" "6#/,&*&%'arcsinG\"\"\"%\"xGF'F'*&%'arccosGF'F(F'F'*&%#PiGF'\"\"#!\"\" " }{TEXT -1 34 " is a consequence of the formula " }{XPPEDIT 18 0 "si n(Pi/2-theta) = cos*theta;" "6#/-%$sinG6#,&*&%#PiG\"\"\"\"\"#!\"\"F*%& thetaGF,*&%$cosGF*F-F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "alpha = arcsin*x;" "6#/%&alphaG*&%'arcsinG\"\" \"%\"xGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta = arccos*x;" "6#/% %betaG*&%'arccosG\"\"\"%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sin*alpha = x;" "6#/*&%$sinG\"\"\"% &alphaGF&%\"xG" }{TEXT -1 7 ", with " }{XPPEDIT 18 0 "-Pi/2<=alpha" "6 #1,$*&%#PiG\"\"\"\"\"#!\"\"F)%&alphaG" }{XPPEDIT 18 0 "``<=Pi/2" "6#1% !G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "cos*be ta = x;" "6#/*&%$cosG\"\"\"%%betaGF&%\"xG" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "0<=beta" "6#1\"\"!%%betaG" }{XPPEDIT 18 0 "``<=Pi" "6#1 %!G%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " } {XPPEDIT 18 0 "sin*alpha = cos*beta;" "6#/*&%$sinG\"\"\"%&alphaGF&*&%$ cosGF&%%betaGF&" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "sin*alpha = sin(Pi/2-beta);" "6#/*&%$sinG\"\"\"%&alphaGF&-F%6#,&*&%#PiGF&\"\"#!\" \"F&%%betaGF." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The ine quality " }{XPPEDIT 18 0 "0<=beta" "6#1\"\"!%%betaG" }{XPPEDIT 18 0 "` `<=Pi" "6#1%!G%#PiG" }{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "-Pi /2<=Pi/2-beta" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F),&*&F&F'F(F)F'%%betaGF) " }{XPPEDIT 18 0 "``<=Pi/2" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "Since the sine function is one-t o-one on the interval " }{XPPEDIT 18 0 "[-Pi/2,Pi/2]" "6#7$,$*&%#PiG\" \"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "sin*alpha = sin(Pi/2-beta);" "6#/*&%$sinG\"\"\"%&alphaGF&-F%6#,& *&%#PiGF&\"\"#!\"\"F&%%betaGF." }{TEXT -1 14 " implies that " } {XPPEDIT 18 0 "alpha=Pi/2-beta" "6#/%&alphaG,&*&%#PiG\"\"\"\"\"#!\"\"F (%%betaGF*" }{TEXT -1 11 ", that is " }{XPPEDIT 18 0 "alpha+beta=Pi/2 " "6#/,&%&alphaG\"\"\"%%betaGF&*&%#PiGF&\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 16 "This shows that " }{XPPEDIT 18 0 "arcsin* x+arccos*x = Pi/2;" "6#/,&*&%'arcsinG\"\"\"%\"xGF'F'*&%'arccosGF'F(F'F '*&%#PiGF'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT -1 43 "It is standard practice to use the f ormula " }{XPPEDIT 18 0 "Int(1/sqrt(1-x^2),x) = arcsin*x+c;" "6#/-%$In tG6$*&\"\"\"F(-%%sqrtG6#,&F(F(*$%\"xG\"\"#!\"\"F0F.,&*&%'arcsinGF(F.F( F(%\"cGF(" }{TEXT -1 11 " involving " }{XPPEDIT 18 0 "arcsin*x;" "6#*& %'arcsinG\"\"\"%\"xGF%" }{TEXT -1 35 " rather than the formula involvi ng " }{XPPEDIT 18 0 "arccos*x;" "6#*&%'arccosG\"\"\"%\"xGF%" }{TEXT -1 2 ". 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F\"F\\an%\"4G/$\"&*4>F\\an%\"6GFabnF^an-%+AXESLABELSG6%%!GF`dn-F`_n6#% (DEFAULTG-%%VIEWG6$;$FdvFh\\nF]`nFgdn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" }}{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 16 "Sin ce the graph " }{XPPEDIT 18 0 "y = tan*x;" "6#/%\"yG*&%$tanG\"\"\"%\"x GF'" }{TEXT -1 14 " has the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG " }{TEXT -1 39 " as a tangent at the origin, the graph " }{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\"%\"xGF'" }{TEXT -1 19 " al so has the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 30 " \+ as a tangent at the origin. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The grap h of " }{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\"%\"xG F'" }{TEXT -1 15 " has the lines " }{XPPEDIT 18 0 "y = Pi/2;" "6#/%\"y G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = -Pi/ 2;" "6#/%\"yG,$*&%#PiG\"\"\"\"\"#!\"\"F*" }{TEXT -1 27 " as horizontal asymptotes. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 554 174 174 {PLOTDATA 2 "6.-%'CURVESG6$7V7$$!33+++++++@!#<$!33ezA;J;@X!#=7$$!3 >++]_>X3?F*$!3ev?]hro*\\%F-7$$!3(***\\(e['zG>F*$!3-].zwYQzWF-7$$!32++v #e:#R=F*$!3#*e\\#[**4XX%F-7$$!33++Dz3/\\')3w:F*$!3syRl\"*ymlVF-7$$!3++](3[L**[ \"F*$!3;H@YJM.IVF-7$$!33+](GrJ3S\"F*$!3e>)pof$y)G%F-7$$!3'***\\P&p:?J \"F*$!3H.^#4E#QUUF-7$$!35++]/vl?7F*$!37(Q%)RLG!)=%F-7$$!3$)***\\-;*=S6 F*$!3zRk(\\CGM8%F-7$$!3#)****\\j3g\\5F*$!3y1[&*e1tiSF-7$$!3[+++DgS'e*F -$!3YES1&=M&zRF-7$$!3A)****\\ON)4()F-$!3A?CUdFq%)QF-7$$!3%****\\(G_#Q \"zF-$!3y())[:VBFy$F-7$$!3E+++&=#HnpF-$!3k\"4*4NP*ej$F-7$$!3Q******RXX lhF-$!3!zc5.[yH[$F-7$$!3$)***\\(QnsK_F-$!3-(\\b\\IA/E$F-7$$!3V*******e /rS%F-$!36lBO)4H*3IF-7$$!3@)**\\(QiE,NF-$!3[^LhOsR^EF-7$$!3%*)**\\i\"R pQEF-$!3$!3c.[6BF'R)))Fir7$$!3yq*\\7)GokYFir$!3#eiO9Z9))Fir7$$\"3`*****\\m+$38F-$\"39U\"Q\\D*HT7F -7$$\"3R***\\(e+M6-d\"F-7$$\"3)3++DBF>e#F-$\"35H?%\\tk #p@F-7$$\"3k,++q*G8[$F-$\"3WHLBcsMUEF-7$$\"3k****\\-\"=7O%F-$\"3wTl]j( *3$*HF-7$$\"3e,+D\"3cD@&F-$\"3%[771+f\\D$F-7$$\"3g,+]d<#y:'F-$\"3y!R) \\(eq8[$F-7$$\"3$))*****4]=2qF-$\"341_!y)*\\Fk$F-7$$\"3H/++v:19zF-$\"3 '*yk^Ujv#y$F-7$$\"3k,+DJl#et)F-$\"35X?Y5rv()QF-7$$\"3'f++]oKUj*F-$\"3a Gx/:BE%)RF-7$$\"3!)**\\7'Gcz/\"F*$\"3#\\7bV_V81%F-7$$\"3B+]iYwJO6F*$\" 3X_z:--iITF-7$$\"3F++]FqqA7F*$\"3s)G2dBJ$*=%F-7$$\"3)***\\7.([JJ\"F*$ \"3M]jkn:,VUF-7$$\"3`+++'ya-S\"F*$\"3=JB;E-])G%F-7$$\"37++vOLL*[\"F*$ \"3plV6o8xHVF-7$$\"3M+]7sUnx:F*$\"3#)p#e.r)GmVF-7$$\"3Y++++]7EP#Q#>F*$\"3CoM^NH1yWF-7$$\"3!3+vy#Gu3?F*$\"3!Ges ?Ue(*\\%F-7$$\"33+++++++@F*$\"33ezA;J;@XF--%'COLOURG6&%$RGBG$\"*++++\" !\")$\"\"!Fa\\lF`\\l-F$6%7$7$F($!3++++++++]F-7$Fe[lFf\\l-Fj[l6&F\\\\lF a\\lFa\\lFa\\l-%*LINESTYLEG6#\"\"$-F$6%7$7$F($\"3++++++++]F-7$Fe[lFc]l Fi\\lF[]l-%%TEXTG6&7$$\"#=!\"\"$\"\"%F\\^lQ-y~=~arctan~x6\"Fi[l-%%FONT G6$%*HELVETICAG\"\"*-Fg]l6&7$$\"#@F\\^l$!\"(!\"#Q\"xF`^lFi\\lFa^l-Fg]l 6&7$F[_l$\"\"'F\\^lQ\"yF`^lFi\\lFa^l-Fg]l6&7$$!#8F]_l$!#WF]_lQ%-p/2F`^ lFi\\l-Fb^l6$%'SYMBOLGFe^l-Fg]l6&7$$F\\^lF\\^l$\"#WF]_lQ$p/2F`^lFi\\lF ]`l-%*AXESTICKSG6$7)/$!&*4>!\"%%#-6G/$!&KF\"F^al%#-4G/$!&iO'!\"&%#-2G/ Fa\\l%\"0G/$\"&iO'Fgal%\"2G/$\"&KF\"F^al%\"4G/$\"&*4>F^al%\"6GFa\\lF]` l-%+AXESLABELSG6%%!GFjbl-Fb^l6#%(DEFAULTG-%%VIEWG6$;$!#@F\\^lFi^l;$!\" 'F\\^lFb_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 47.000000 44.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C urve 8" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "A \"nice\" feature of the inverse tangent functio n is that it is " }{TEXT 259 28 "defined for all real numbers" }{TEXT -1 86 ", unlike the inverse sine and cosine functions which are only d efined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "The domain of the a rctangent function is the set of all real numbers |R, and its range is the open interval" }{XPPEDIT 18 0 "``(-Pi/2,Pi/2) =``" "6#/-%!G6$,$*& %#PiG\"\"\"\"\"#!\"\"F,*&F)F*F+F,F%" }{TEXT -1 1 "\{" }{XPPEDIT 18 0 " y*`|`-Pi/2 < y" "6#2,&*&%\"yG\"\"\"%\"|grGF'F'*&%#PiGF'\"\"#!\"\"F,F& " }{TEXT -1 1 " " }{XPPEDIT 18 0 "``< Pi/2" "6#2%!G*&%#PiG\"\"\"\"\"#! \"\"" }{TEXT -1 2 "\}." }}{PARA 0 "" 0 "" {TEXT -1 58 "The arctangent \+ function has the following special values. " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "arctan*0 = 0;" "6#/*&%'arctanG\"\"\"\"\"!F&F'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(1/ sqrt(3)) = Pi/6;" "6#/-%'arctanG6#*&\"\"\"F(-%%sqrtG6#\"\"$!\"\"*&%#Pi GF(\"\"'F-" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "arctan(-1/sqrt(3 )) = -Pi/6;" "6#/-%'arctanG6#,$*&\"\"\"F)-%%sqrtG6#\"\"$!\"\"F.,$*&%#P iGF)\"\"'F.F." }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arctan*1 = Pi/4;" "6#/*&%'arctanG\"\"\"F&F&*&%#PiGF&\" \"%!\"\"" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "arctan(-1) = -Pi/4 ;" "6#/-%'arctanG6#,$\"\"\"!\"\",$*&%#PiGF(\"\"%F)F)" }{TEXT -1 2 ", \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(sqrt(3)) = Pi/3;" "6#/-%'arctanG6#-%%sqrtG6#\"\"$*&%#PiG\"\"\"F*!\"\"" }{TEXT -1 10 ", " }{XPPEDIT 18 0 "arctan(-sqrt(3)) = -Pi/3;" "6#/-%'a rctanG6#,$-%%sqrtG6#\"\"$!\"\",$*&%#PiG\"\"\"F+F,F," }{TEXT -1 3 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 4 " As " }{XPPEDIT 18 0 "x->infinity" " 6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "arctan(x)->Pi/2" "6#f*6#-%'arctanG6#%\"xG7\"6$%)op eratorG%&arrowG6\"*&%#PiG\"\"\"\"\"#!\"\"F-F-F-" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 4 " As " }{XPPEDIT 18 0 "proc (x) options o perator, arrow; -infinity end proc;" "6#f*6#%\"xG7\"6$%)operatorG%&arr owG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "arcta n(x)->-Pi/2" "6#f*6#-%'arctanG6#%\"xG7\"6$%)operatorG%&arrowG6\",$*&%# PiG\"\"\"\"\"#!\"\"F3F-F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The arctangent function is an \+ " }{TEXT 259 12 "odd function" }{TEXT -1 38 ", that is, it satisfies t he relation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arct an(-x) = -arctan*x;" "6#/-%'arctanG6#,$%\"xG!\"\",$*&F%\"\"\"F(F,F)" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 288 13 "____ _________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "derivative" }{TEXT -1 40 " of \+ the arctangent function is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[arctan*x] = 1/(1+x^2);" "6#/7#*&%'arctanG\"\"\"%\"xGF' *&F'F',&F'F'*$F(\"\"#F'!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 276 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We can check th is by implicit differentiation. " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given \+ " }{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\"%\"xGF'" } {TEXT -1 12 ", we have " }{XPPEDIT 18 0 "tan*y = x;" "6#/*&%$tanG\" \"\"%\"yGF&%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-Pi/2 < y;" "6#2 ,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"yG" }{XPPEDIT 18 0 "`` < Pi/2;" "6#2%!G* &%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "sec^2*y;" "6#*&%$secG\"\"#%\"yG\"\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1;" "6#/*&%#dyG\"\"\"%#dxG!\"\" F&" }{TEXT -1 12 ", so that " }{XPPEDIT 18 0 "dy/dx = 1/(sec^2*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*&%$secG\"\"#%\"yGF&F(" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "sec^2*y = 1+ tan^2*y;" "6#/*&%$secG\"\"#%\"yG\"\"\",&F(F(*&%$tanGF&F'F(F(" }{TEXT -1 16 ", we see that " }{XPPEDIT 18 0 "dy/dx = 1/(1+tan^2*y);" "6#/* &%#dyG\"\"\"%#dxG!\"\"*&F&F&,&F&F&*&%$tanG\"\"#%\"yGF&F&F(" }{TEXT -1 13 ", that is, " }{XPPEDIT 18 0 "dy/dx = 1/(1+x^2);" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&F&F&,&F&F&*$%\"xG\"\"#F&F(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The corresponding \+ integration formula is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/(1+x^2),x) = arctan*x+c;" "6#/-%$IntG6$*&\"\"\"F( ,&F(F(*$%\"xG\"\"#F(!\"\"F+,&*&%'arctanGF(F+F(F(%\"cGF(" }{TEXT -1 2 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 277 15 "_______________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Maple \"knows\" these res ults." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Diff(arctan(x),x);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%'arctanG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&,&F&F&*$)%\"xG\"\"#F&F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int( 1/(1+x^2),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*&\"\"\"F',&F'F'*$)%\"xG\"\"#F'F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%'arctanG6#%\"xG\"\"\"%\"cGF*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "A definite integral involving arc tan" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "The following picture shows the graph of " }{XPPEDIT 18 0 "f(x) = arctan*x;" "6#/-%\"fG6#%\"xG*&%'arctanG\"\"\"F'F*" }{TEXT -1 30 " together with its derivative " }{XPPEDIT 18 0 "`f '`(x) = 1/(1 +x^2);" "6#/-%$f~'G6#%\"xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Int(1/(1+x^2),x=0..1);\nvalue(%);\nevalf(evalf(%,15)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&F'F'*$)%\"xG \"\"#F'F'!\"\"/F+;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"% !\"\"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M;)R&y!#5" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "This area can be compared with the area of the trapezoid with vertices at the points" } {XPPEDIT 18 0 "``(0,0),``(0,1),``(1,1/2);" "6%-%!G6$\"\"!F&-F$6$F&\"\" \"-F$6$F)*&F)F)\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(1,0) " "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 "which is outlined in " }{TEXT 256 4 "blue" }{TEXT -1 27 " in the f ollowing picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 305 "f := x -> 1/(1+x^2):\np1 := plot(f(x),x=0.. 2,y=0..1.2,color=red,thickness=2):\na := 0: b := 1:\np2 := plot(f(x),x =a..b,color=COLOR(RGB,.87,.87,.93),filled=true):\np3 := plot([[b,0],[b ,f(b)]],color=black):\np4 := plot([[0,0],[a,f(a)],[b,f(b)],[b,0],[0,0] ],color=blue,thickness=2):\nplots[display]([p1,p2,p3,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 411 278 278 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"\"! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "Two standard integrals involving inverse trigonometric functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 " 1: " }{XPPEDIT 18 0 "Int(1/(a^2+x^2),x) = 1/a" "6#/-%$ IntG6$*&\"\"\"F(,&*$%\"aG\"\"#F(*$%\"xGF,F(!\"\"F.*&F(F(F+F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c;" "6#,&-%'arctanG6#*&%\"xG\"\" \"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "The integration for mula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(a^2+x^ 2),x) = 1/a;" "6#/-%$IntG6$*&\"\"\"F(,&*$%\"aG\"\"#F(*$%\"xGF,F(!\"\"F .*&F(F(F+F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c" "6#,&-%'ar ctanG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" } {TEXT -1 36 ", can be checked by differentiation." }}{PARA 0 "" 0 "" {TEXT -1 28 "We have (by the chain rule) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arctan(x/a)];" "6#7#-%'arctanG6#*&%\"xG\"\"\"% \"aG!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "1/``(1+x^2/(a^2));" "6# *&\"\"\"F$-%!G6#,&F$F$*&%\"xG\"\"#*$%\"aGF+!\"\"F$F." }{TEXT -1 1 " " }{TEXT 265 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x/a]" "6#7#*&%\"xG\"\" \"%\"aG!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= " } {XPPEDIT 18 0 "1/``(1+x^2/(a^2));" "6#*&\"\"\"F$-%!G6#,&F$F$*&%\"xG\" \"#*$%\"aGF+!\"\"F$F." }{TEXT -1 1 " " }{TEXT 266 1 "." }{XPPEDIT 18 0 "1/a" "6#*&\"\"\"F$%\"aG!\"\"" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "1/(a*`.`*``(1+x^2/(a^2)));" "6#*&\"\" \"F$*(%\"aGF$%\".GF$-%!G6#,&F$F$*&%\"xG\"\"#*$F&F.!\"\"F$F$F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "a/(a^2*` .`*``(1+x^2/(a^2)));" "6#*&%\"aG\"\"\"*(F$\"\"#%\".GF%-%!G6#,&F%F%*&% \"xGF'*$F$F'!\"\"F%F%F0" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "a/(a^2+x^2)" "6#*&%\"aG\"\"\",&*$F$\"\"#F% *$%\"xGF(F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(a/(a^2+x^2), x) = arctan(x/a)+c" "6#/-%$IntG6$*&%\"aG\"\"\",&*$F(\"\"#F)*$%\"xGF,F) !\"\"F.,&-%'arctanG6#*&F.F)F(F/F)%\"cGF)" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 41 "which is equivalent to the stated result." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We can us e Maple's integration procedure to obtain this formula. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a := \+ 'a': x := 'x':\nInt(1/(a^2+x^2),x);\n``=value(%)+c;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&*$)%\"aG\"\"#F'F'*$)%\"xGF,F'F'! \"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"aG!\"\"-%'arctanG 6#*&%\"xG\"\"\"F'F(F.F.%\"cGF." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 " 2: " }{XPPEDIT 18 0 "Int(1/sqrt(a^2-x^2),x) = arcsin(x/ a)+c" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&*$%\"aG\"\"#F(*$%\"xGF/!\"\"F 2F1,&-%'arcsinG6#*&F1F(F.F2F(%\"cGF(" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "The int egration formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(1/sqrt(a^2-x^2),x) = arcsin(x/a)+c" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG 6#,&*$%\"aG\"\"#F(*$%\"xGF/!\"\"F2F1,&-%'arcsinG6#*&F1F(F.F2F(%\"cGF( " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 36 ", can be checked by differentiat ion." }}{PARA 0 "" 0 "" {TEXT -1 28 "We have (by the chain rule) " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arcsin(x/a)];" "6#7#-%' arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/ sqrt(1-x^2/a^2)" "6#*&\"\"\"F$-%%sqrtG6#,&F$F$*&%\"xG\"\"#*$%\"aGF+! \"\"F.F." }{TEXT -1 1 " " }{TEXT 263 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x/a]" "6#7#*&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "1/ sqrt(1-x^2/a^2)" "6#*&\"\"\" F$-%%sqrtG6#,&F$F$*&%\"xG\"\"#*$%\"aGF+!\"\"F.F." }{TEXT -1 1 " " } {TEXT 264 1 "." }{XPPEDIT 18 0 "1/a" "6#*&\"\"\"F$%\"aG!\"\"" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "1/ (a*`. `*sqrt(1-x^2/a^2))" "6#*&\"\"\"F$*(%\"aGF$%\".GF$-%%sqrtG6#,&F$F$*&%\" xG\"\"#*$F&F.!\"\"F0F$F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "1/ sqrt(a^2-x^2)" "6#*&\"\"\"F$-%%sqrtG6#,&* $%\"aG\"\"#F$*$%\"xGF+!\"\"F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 67 "Maple gives what, at first sight, appears to be a different result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(1/sqrt(a^2-x^2),x);\n``=value(% )+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,&*$)%\"a G\"\"#F'F'*$)%\"xGF-F'!\"\"#F'F-F1F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,&-%'arctanG6#*&%\"xG\"\"\",&*$)%\"aG\"\"#F+F+*$)F*F0F+!\"\"#F3F 0F+%\"cGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 212 128 128 {PLOTDATA 2 "6+-%'CU RVESG6$7&7$$\"\"!F)F(7$$\"\"#F)F(7$F+$\"\"\"F)F'-%&COLORG6&%$RGBG$\"#X 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\\adF?$\"3^`ajs;zB?F?7$$\"3\\L1]`QONdF?$\"3K:%[sd-u2#F?7$$\"3EIj+zu#[r &F?$\"3jDgCk)\\3l(=#F?7$$\"3qD*QXj)4sc F?$\"36ae@jBRWAF?7$$\"3=I]E=]D]cF?$\"3cth(>vH))H#F?7$$\"3![jo)ekPFcF?$ \"3![&ows*zUN#F?7$$\"3mUZ1?*\\Tg&F?$\"3!QiUl@Y!4CF?7$$\"3'>7t,?PBe&F?$ \"3Qgq)*R)o\"fCF?7$$\"3w_*pi:'ycbF?$\"3K<'yZ\">P;DF?7$$\"3)e%*QVLOM`&F ?$\"3w/8kI.JnDF?7$$\"3Oa(*)>FE!3bF?$\"33THnrDQ@EF?7$$\"3Q4&Hno5K[&F?$ \"37>)e8:,Hn#F?7$$\"3=&o,^'e+caF?$\"3KX`aKH+GFF?-%'COLOURG6&F3F)F)F)-F $6$7%7$$\"3!***************=!# " 0 "" {MPLTEXT 1 0 65 "plot(arctan(x/sqrt(1-x^2)),x =-1..1,numpoints=80,tickmarks=[2,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 216 262 262 {PLOTDATA 2 "6&-%'CURVESG6$7\\p7$$!1+++!>B^t*!#;$!1\"*3ug9 7S8!#:7$$!1+++U#*))>'*F*$!1TL'>D%>%H\"F-7$$!1+++%HbY]*F*$!1s;.#=IZD\"F -7$$!1+++S+ZX#*F*$!1q\\$37X)z6F-7$$!1******Qnc%)*)F*$!1(=^HsTi6\"F-7$$ !1+++mM!\\s)F*$!1cW7OfFg5F-7$$!1+++sB;%[)F*$!1t=Rkg)H,\"F-7$$!1+++9*)) [B)F*$!1>)['[\\Lv'*F*7$$!1+++Z**3xzF*$!1M;9t`'[B*F*7$$!1+++Wx6?xF*$!1B 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D^]QFjv$\"11&zcik9&QFjv7$$\"1++++*)[1kFjv$\"1qF@$Qz3T'Fjv7$$\"1++++e;b ()Fjv$\"17mS5(*Qm()Fjv7$$\"1+++]9qW6F*$\"1#RTIA;s9\"F*7$$\"1+++?OY&Q\" F*$\"1$)oz*pM**Q\"F*7$$\"1++++j;U;F*$\"1nGkv!Q'\\;F*7$$\"1+++?N'y)=F*$ \"1vWe\"Gh#**=F*7$$\"1+++5(y1;#F*$\"1@#Q)QS&y<#F*7$$\"1+++5m\"pR#F*$\" 1A,U@.%F*7$$\" 1+++g.LxTF*$\"19+&4%z[4VF*7$$\"1++++*f*HWF*$\"1Zuh5rP*e%F*7$$\"1+++!zd En%F*$\"1xT>.k&>'[F*7$$\"1+++]1-N\\F*$\"1Z;6&=<6;&F*7$$\"1+++q92&=&F*$ \"1)HrUZT5X&F*7$$\"1+++!=Q#[aF*$\"1U@tt**yhdF*7$$\"1+++SuY'o&F*$\"1(yK c&yf[gF*7$$\"1+++]`v\\fF*$\"1uu`&4_BP'F*7$$\"1++++$R>?'F*$\"1@N5t*)*)* o'F*7$$\"1+++I%4NX'F*$\"1YG;?h#[,(F*7$$\"1,++!Rvir'F*$\"1dI&QWLSO(F*7$ $\"1+++]CHepF*$\"1U#G.gQdp(F*7$$\"1+++55:1sF*$\"1=,mF2*o/)F*7$$\"1+++? j\")zuF*$\"1Brl'\\e,X)F*7$$\"1+++SyfFxF*$\"1!\\RX'*y<$))F*7$$\"1+++]3+ \")zF*$\"12GsUMNT#*F*7$$\"1+++!>0(Q#)F*$\"1upQQ\\1#o*F*7$$\"1+++Icev%) F*$\"1C9ofyO65F-7$$\"1++++D4G()F*$\"1f-6j*G41\"F-7$$\"1******psny*)F*$ \"1&p(y$=-\\6\"F-7$$\"1+++!4DnC*F*$\"149=.YzC\"F-7$$\"1******4W=c(*F*$\"1Z=[_9_\\8F-7$%%FAILGF`dl-%'COLOUR G6&%$RGBG$\"#5!\"\"\"\"!Fhdl-%+AXESLABELSG6$Q\"x6\"%!G-%*AXESTICKSG6$ \"\"#\"\"$-%%VIEWG6$;$FgdlFhdl$\"\"\"Fhdl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "It would be a nice little differentiation exercis e to check by hand that " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG !\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "[arctan(x/sqrt(1-x^2))];" "6#7 #-%'arctanG6#*&%\"xG\"\"\"-%%sqrtG6#,&F)F)*$F(\"\"#!\"\"F0" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/sqrt(1-x^2);" "6#*&\"\"\"F$-%%sqrtG6#,&F$F$ *$%\"xG\"\"#!\"\"F," }{TEXT -1 50 ", without first simplifying the lef t hand side to " }{XPPEDIT 18 0 "arcsin*x;" "6#*&%'arcsinG\"\"\"%\"xGF %" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Diff(arctan(x/sqrt(1-x^2)),x);\nvalue(%);\nsi mplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%'arctanG6#*& %\"xG\"\"\",&F+F+*$)F*\"\"#F+!\"\"#F0F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"\"F&*$,&F&F&*$)%\"xG\"\"#F&!\"\"#F&F,F-F&*&F+ F,F(#!\"$F,F&F&,&F&F&*&F+F,F(F-F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&\"\"\"F$*$,&F$F$*$)%\"xG\"\"#F$!\"\"#F$F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 64 "Examples of integrals involving inverse trigonometric f unctions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 290 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "Find (a) \+ " }{XPPEDIT 18 0 "Int(1/(x^2+9),x);" "6#-%$IntG6$*&\"\"\"F',&*$%\"xG \"\"#F'\"\"*F'!\"\"F*" }{TEXT -1 13 " and (b) " }{XPPEDIT 18 0 "In t(1/(x^2+9),x=-sqrt(3)..sqrt(3))" "6#-%$IntG6$*&\"\"\"F',&*$%\"xG\"\"# F'\"\"*F'!\"\"/F*;,$-%%sqrtG6#\"\"$F--F26#F4" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 291 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT 293 8 "Method I" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 22 "The standard formula \+ " }{XPPEDIT 18 0 "Int(1/(a^2+x^2),x) = 1/a" "6#/-%$IntG6$*&\"\"\"F(,&* $%\"aG\"\"#F(*$%\"xGF,F(!\"\"F.*&F(F(F+F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c;" "6#,&-%'arctanG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF )" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "a=3" "6#/%\"aG\"\"$" }{TEXT -1 8 " gives " }{XPPEDIT 18 0 "Int(1/(x^2+9),x)=1/3" "6#/-%$IntG6$*& \"\"\"F(,&*$%\"xG\"\"#F(\"\"*F(!\"\"F+*&F(F(\"\"$F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(x/3) + c" "6#,&-%'arctanG6#*&%\"xG\"\"\"\"\"$!\" \"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 294 9 "Method II " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "We make the " } {TEXT 259 20 "inverse substitution" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x \+ = 3*tan*theta;" "6#/%\"xG*(\"\"$\"\"\"%$tanGF'%&thetaGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "-Pi/2 \+ " 0 "" {MPLTEXT 1 0 65 "Int(1/(x^2+9),x=-sqrt(3)..sqrt(3));\nvalue(%); \nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\" \"\"F',&*$)%\"xG\"\"#F'F'\"\"*F'!\"\"/F+;,$*$\"\"$#F'F,F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"*!\"\"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/&e1\\$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 282 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "Find (a) " }{XPPEDIT 18 0 "Int(1/sqrt(4-x^2),x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"%F' *$%\"xG\"\"#!\"\"F0F." }{TEXT -1 13 " and (b) " }{XPPEDIT 18 0 "In t(1/sqrt(4-x^2),x = -1 .. 1);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"% F'*$%\"xG\"\"#!\"\"F0/F.;,$F'F0F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 0 "" 0 "" {TEXT 285 8 "Method I" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 22 "The standard formula " }{XPPEDIT 18 0 "I nt(1/sqrt(a^2-x^2),x) = arcsin(x/a)+c;" "6#/-%$IntG6$*&\"\"\"F(-%%sqrt G6#,&*$%\"aG\"\"#F(*$%\"xGF/!\"\"F2F1,&-%'arcsinG6#*&F1F(F.F2F(%\"cGF( " }{TEXT -1 6 " with " }{XPPEDIT 18 0 "a = 2;" "6#/%\"aG\"\"#" }{TEXT -1 8 " gives " }{XPPEDIT 18 0 "Int(1/sqrt(4-x^2),x) = arcsin(x/2)+c; " "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&\"\"%F(*$%\"xG\"\"#!\"\"F1F/,&-%' arcsinG6#*&F/F(F0F1F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 289 9 "Method II" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "We make the " }{TEXT 259 20 "inverse substitution" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x = 2*sin*theta;" "6#/%\"xG*(\"\"#\"\"\"%$sinGF'%&th etaGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "-Pi/2 <= theta;" "6#1,$ *&%#PiG\"\"\"\"\"#!\"\"F)%&thetaG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%! G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 18 ", that is, we let " }{XPPEDIT 18 0 "theta = arcsin(x/2);" "6#/%&thetaG-%'arcsinG6#*&%\"xG\"\"\"\"\"# !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/sqrt(4-x^2),x);" "6#-%$In tG6$*&\"\"\"F'-%%sqrtG6#,&\"\"%F'*$%\"xG\"\"#!\"\"F0F." }{TEXT -1 8 " \+ ... " }{XPPEDIT 18 0 "PIECEWISE([x = 2*sin*theta, theta = arcsin(x/ 2)],[dx = 2*cos*theta*`.`*d*theta, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*( \"\"#\"\"\"%$sinGF+%&thetaGF+/F--%'arcsinG6#*&F(F+F*!\"\"7$/%#dxG*.F*F +%$cosGF+F-F+%\".GF+%\"dGF+F-F+%!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ Int(2*cos*theta/sqrt(4-4*sin^2*theta),theta);" "6#/%!G-%$IntG6$**\"\"# \"\"\"%$cosGF*%&thetaGF*-%%sqrtG6#,&\"\"%F**(F1F**$%$sinGF)F*F,F*!\"\" F5F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Int(2*cos*theta/(2*cos*theta),theta);" "6#/%!G-%$IntG6$**\" \"#\"\"\"%$cosGF*%&thetaGF**(F)F*F+F*F,F*!\"\"F," }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Int(1,theta);" "6#/%!G-%$IntG6$\"\"\"%&thetaG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = theta+c;" "6#/%!G,&%&thetaG\"\"\" %\"cGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = arcsin(x/2)+c;" "6#/%!G,&-%'arcsinG6#*&%\"xG\"\"\" \"\"#!\"\"F+%\"cGF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(1/sqrt(4-x^2),x = -1 .. 1) = arcsin(x/2);" "6#/- %$IntG6$*&\"\"\"F(-%%sqrtG6#,&\"\"%F(*$%\"xG\"\"#!\"\"F1/F/;,$F(F1F(-% 'arcsinG6#*&F/F(F0F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, `` ],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$,$F'!\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=arcsin(1/2)-arcsin(-1/2)" "6#/%!G,&-%'arcsinG6# *&\"\"\"F*\"\"#!\"\"F*-F'6#,$*&F*F*F+F,F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi/6+Pi/6" "6#/%!G,&*&%# PiG\"\"\"\"\"'!\"\"F(*&F'F(F)F*F(" }{XPPEDIT 18 0 "``=Pi/3" "6#/%!G*&% #PiG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 295 1 "~" }{TEXT -1 14 " \+ 1.047197551. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Int(1/sqrt(4-x^2),x=-1..1);\nvalue(%);\nevalf(ev alf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,& \"\"%F'*$)%\"xG\"\"#F'!\"\"#F'F.F//F-;F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^v>Z5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Find " }{XPPEDIT 18 0 "Int(1/(x^2+4*x+29),x);" "6#-%$IntG6$*&\"\"\"F',(*$%\"xG\"\"#F'*&\"\"% F'F*F'F'\"#HF'!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "By completi ng the square we see that " }{XPPEDIT 18 0 "x^2+4*x+29 = ``(x^2+4*x+4 )+25;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(F(\"#HF(,&-%!G6#,(*$F&F'F( *&F*F(F&F(F(F*F(F(\"#DF(" }{XPPEDIT 18 0 "`` = (x+2)^2+25;" "6#/%!G,&* $,&%\"xG\"\"\"\"\"#F)F*F)\"#DF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Consider the substitution " }{XPPEDIT 18 0 "u=x+2" "6#/% \"uG,&%\"xG\"\"\"\"\"#F'" }{TEXT -1 24 " in the given integral. " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^2+4*x+29),x) = Int(1/((x+2)^2+25),x);" "6#/-%$IntG6$*&\"\"\"F(,(*$%\"xG\"\"#F(*&\" \"%F(F+F(F(\"#HF(!\"\"F+-F%6$*&F(F(,&*$,&F+F(F,F(F,F(\"#DF(F0F+" } {TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=x+2,``],[du=dx,``]) " "6#-%*PIECEWISEG6$7$/%\"uG,&%\"xG\"\"\"\"\"#F+%!G7$/%#duG%#dxGF-" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=I nt(1/(u^2+25),u)" "6#/%!G-%$IntG6$*&\"\"\"F),&*$%\"uG\"\"#F)\"#DF)!\" \"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/5" "6#/%!G*&\"\"\"F&\"\"&!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(u/5) +c" "6#,&-%'arctanG6#* &%\"uG\"\"\"\"\"&!\"\"F)%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1 /5" "6#/%!G*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcta n((x+2)/5)+c" "6#,&-%'arctanG6#*&,&%\"xG\"\"\"\"\"#F*F*\"\"&!\"\"F*%\" cGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(1/(x^ 2+4*x+29),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*&\"\"\"F',(*$)%\"xG\"\"#F'F'*&\"\"%F'F+F'F'\"#HF'!\"\"F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"&F(-%'arctanG6#,&*& F)!\"\"%\"xGF(F(#\"\"#F)F(F(F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT 298 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Find " }{XPPEDIT 18 0 "Int(1/sqrt(9-5*x^2),x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"* F'*&\"\"&F'*$%\"xG\"\"#F'!\"\"F2F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT 299 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(9-5*x^2),x) = Int(1/(sqrt(5)*sqrt( 9/5-x^2)),x);" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&\"\"*F(*&\"\"&F(*$% \"xG\"\"#F(!\"\"F3F1-F%6$*&F(F(*&-F*6#F/F(-F*6#,&*&F-F(F/F3F(*$F1F2F3F (F3F1" }{XPPEDIT 18 0 "`` = 1/sqrt(5);" "6#/%!G*&\"\"\"F&-%%sqrtG6#\" \"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(9/5-x^2),x);" "6# -%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*&\"\"*F'\"\"&!\"\"F'*$%\"xG\"\"#F/F/F1 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/sqrt(5);" "6#/%!G*&\"\"\"F&-%%sq rtG6#\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(sqrt(5)*x/3)+c ;" "6#,&-%'arcsinG6#*(-%%sqrtG6#\"\"&\"\"\"%\"xGF,\"\"$!\"\"F,%\"cGF, " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(1/sqrt(9-5*x^2),x);\n``=value(%)+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,&\"\"*F'*&\"\"&F ')%\"xG\"\"#F'!\"\"#F'F/F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,& *&#\"\"\"\"\"&F(*&F)#F(\"\"#-%'arcsinG6#,$*(\"\"$!\"\"F)F+%\"xGF(F(F(F (F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example \+ 5 " }}{PARA 0 "" 0 "" {TEXT 300 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Find " }{XPPEDIT 18 0 "Int(1/sqrt(3-4*x-4*x^2) ,x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,(\"\"$F'*&\"\"%F'%\"xGF'!\"\"*& F.F'*$F/\"\"#F'F0F0F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 301 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "By completi ng the square we see that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "3-4*x-4*x^2 = -4(x^2+1)+3;" "6#/,(\"\"$\"\"\"*&\"\"%F&% \"xGF&!\"\"*&F(F&*$F)\"\"#F&F*,&-F(6#,&*$F)F-F&F&F&F*F%F&" }{XPPEDIT 18 0 "`` = -4*(x^2+x+1/4)+3+1;" "6#/%!G,(*&\"\"%\"\"\",(*$%\"xG\"\"#F( F+F(*&F(F(F'!\"\"F(F(F.\"\"$F(F(F(" }{XPPEDIT 18 0 "``=-4(x+1/2)^2+4" "6#/%!G,&*$-\"\"%6#,&%\"xG\"\"\"*&F,F,\"\"#!\"\"F,F.F/F(F," }{XPPEDIT 18 0 "`` = 4-(2*x+1)^2;" "6#/%!G,&\"\"%\"\"\"*$,&*&\"\"#F'%\"xGF'F'F'F 'F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Consider the substitution " }{XPPEDIT 18 0 "u = 2*x+1;" "6#/%\"uG,&*&\"\"#\"\"\"% \"xGF(F(F(F(" }{TEXT -1 24 " in the given integral. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(3-4*x-4*x^2),x) = Int(1 /sqrt(4-(2*x+1)^2),x);" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,(\"\"$F(*&\" \"%F(%\"xGF(!\"\"*&F/F(*$F0\"\"#F(F1F1F0-F%6$*&F(F(-F*6#,&F/F(*$,&*&F4 F(F0F(F(F(F(F4F1F1F0" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWIS E([u = 2*x+1, ``],[du = 2*`.`*dx, ``],[``(1/2)*du = dx, ``]);" "6#-%*P IECEWISEG6%7$/%\"uG,&*&\"\"#\"\"\"%\"xGF,F,F,F,%!G7$/%#duG*(F+F,%\".GF ,%#dxGF,F.7$/*&-F.6#*&F,F,F+!\"\"F,F1F,F4F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F& \"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(4-u^2),u)" "6#- %$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"%F'*$%\"uG\"\"#!\"\"F0F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(u/2)+c;" "6#,&-%'arcsinG6#*&%\"uG\"\"\"\"\"# !\"\"F)%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\" \"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin((2*x+1)/2)+c; " "6#,&-%'arcsinG6#*&,&*&\"\"#\"\"\"%\"xGF+F+F+F+F+F*!\"\"F+%\"cGF+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(1/sqrt(3-4*x-4*x^2),x);\n``=value(%)+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,(\"\"$F'*&\"\"%F '%\"xGF'!\"\"*&F,F')F-\"\"#F'F.#F'F1F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(-%'arcsinG6#,&%\"xGF(F'F(F(F(%\"cGF(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 0 " " 0 "" {TEXT 280 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 49 "Find the area of the region bounded by the curve " } {XPPEDIT 18 0 "y=(2*x+1)/(x^2+1)" "6#/%\"yG*&,&*&\"\"#\"\"\"%\"xGF)F)F )F)F),&*$F*F(F)F)F)!\"\"" }{TEXT -1 6 ", the " }{TEXT 302 1 "x" } {TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"% " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "f := x -> (2*x+1)/(1+x^2):\np1 := plot(f(x),x=-.6..5 ,y=0..2,color=red,thickness=2):\na := -1/2: b := 4:\np2 := plot(f(x),x =a..b,color=COLOR(RGB,.87,.87,.93),filled=true):\np3 := plot([[b,0],[b ,f(b)]],color=black):\nplots[display]([p1,p2,p3]);" }}{PARA 13 "" 1 " " {GLPLOT2D 475 193 193 {PLOTDATA 2 "6'-%'CURVESG6%7]o7$$!3w********** ****f!#=$!3U7$$!3smmmm$f$ zZF*$\"3a61hh*eAf$F27$$!3m****\\AHK[UF*$\"3!3Jx[B3NF\"F*7$$!3:LLLykG

F*$\"39b'***)\\\"F_o7$$\"3)fmm;h]V\"RF*$\"3iE?dy6*fa\"F_o7$$\"3v**** *\\Rdk]%F*$\"35'QRoC^.e\"F_o7$$\"3]mm\"zp%)4\"[F*$\"3#)*GwzB'R$f\"F_o7 $$\"3CLL$3+7b6&F*$\"3'pt[B%H\\.;F_o7$$\"3V***\\PIR+U&F*$\"3JyUR'f/3h\" F_o7$$\"3smmm1mcCdF*$\"3h*e2yX-bh\"F_o7$$\"3!RLL3U%z#*fF*$\"3zS/0b_h<; F_o7$$\"3++++NA-hiF*$\"3MY::A$ezh\"F_o7$$\"32mm;\\+DHlF*$\"3\"yKem'Hl; ;F_o7$$\"3ELLLjyZ(z'F*$\"3d'3TI3>Qh\"F_o7$$\"3)fmm;%)*R,uF*$\"3f!4koiZ Cg\"F_o7$$\"3q)*****>=K0!)F*$\"3E!G+ya\">&e\"F_o7$$\"3_)******HD\"=#*F *$\"3QtXtg?JP:F_o7$$\"37+++=')oQ5F_o$\"3I:HJ9'4.[\"F_o7$$\"3ALL$GIB[9 \"F_o$\"317&**e22PU\"F_o7$$\"3Ommm3x-r7F_o$\"3iv5kQ'\\UN\"F_o7$$\"3[mm m%RRzP\"F_o$\"3)y(eFlQq&H\"F_o7$$\"3Y****\\,JI-:F_o$\"3$y4lU_q&H7F_o7$ $\"3=mmmagQ7;F_o$\"3-v$>Qm;O<\"F_o7$$\"3o****\\,X;LF_o$\"3X)R,'*=_G, \"F_o7$$\"3amm;eDPy?F_o$\"35%GFEd*z$p*F*7$$\"35LLL)*eB(>#F_o$\"3q@r!=# RPc#*F*7$$\"3(GLLG],2K#F_o$\"3`OK\\UK\\M))F*7$$\"3O****\\u'y\"GCF_o$\" 3+$=,/\"RL#\\)F*7$$\"3/LLLkpDWDF_o$\"3Y.'HJc$3Z\")F*7$$\"3O*****f>xTm# F_o$\"3Y%[#\\Pq'[\"yF*7$$\"3C*****pu&\\\"y#F_o$\"3fJZSKR'>^(F*7$$\"3k* ***\\xu+&*GF_o$\"3UeV-tv*zB(F*7$$\"3]*****4!H/@IF_o$\"3O#)>[X(yR&pF*7$ $\"3llmmM8HMJF_o$\"3W(*QS8:Q:nF*7$$\"3s******4#3_D$F_o$\"3u*fh,w!4:PF_o$\"3e'R\"p N&Q`p&F*7$$\"3;mmmOgFIQF_o$\"3gpnQS#yk_&F*7$$\"37****\\P\\'3&RF_o$\"3; &=\"[\"Qu%f`F*7$$\"3#RLL8Q1q1%F_o$\"3%e``[Mst?&F*7$$\"33LLL#yxd=%F_o$ \"3EdAr$>M+1&F*7$$\"3Smm;'plNI%F_o$\"3H/!fuH::#\\F*7$$\"3@+++c0!=T%F_o $\"3QX.*eGM/![F*7$$\"31nmm>*\\e`%F_o$\"3u3.T(fd%oYF*7$$\"3ALLL\\6!ok%F _o$\"39Jk%yn]hb%F*7$$\"39++],$)4lZF_o$\"3*pZW:0m>W%F*7$$\"3w****\\qPKy [F_o$\"3K?(=i?+xL%F*7$$\"\"&\"\"!$\"3tI#p2Bp2B%F*-%'COLOURG6&%$RGBG$\" *++++\"!\")$Fg_lFg_lFa`l-%*THICKNESSG6#\"\"#-%)POLYGONSG6]o7&7$$!3++++ ++++]F*Fg_l7$F[alFa`l7$$!3;++DJSc4XF*$\"33uGXU$)4^\")F27$F_alFg_l7&Fca lF^al7$$!3K++]i!G\">SF*$\"3i!e\")\\>D*)o\"F*7$FfalFg_l7&FjalFeal7$$!3/ +vV[BS#f$F*$\"39^\\(Ho6M\\#F*7$F]blFg_l7&FablF\\bl7$$!3!)**\\PMmnlJF*$ \"31jj@$R\"[MLF*7$FdblFg_l7&FhblFcbl7$$!3w*\\i!R*ydo#F*$\"3HX?'RjOqJ%F *7$F[clFg_l7&F_clFjbl7$$!3s***\\PC\")e?#F*$\"3ya4!\\;O*G`F*7$FbclFg_l7 &FfclFacl7$$!3$)*****\\Z-Gs\"F*$\"3-+j#3(\\YljF*7$FiclFg_l7&F]dlFhcl7$ $!3&****\\iqB(R7F*$\"3qp&GGkPrF\"F_o7$FjflFg_l7&F^glFifl7$$\"3x**\\i!*yv8?F *$\"3Kfgy-Q3[8F_o7$FaglFg_l7&FeglF`gl7$$\"35+]i!z)3\"\\#F*$\"3X\"*)Rm \"yn59F_o7$FhglFg_l7&F\\hlFggl7$$\"3>**\\7y*)oUMF*$\"3NK2[Mkh4:F_o7$F_ hlFg_l7&FchlF^hl7$$\"39,+]Pn_@WF*$\"3!R*R'4Tmhd\"F_o7$FfhlFg_l7&FjhlFe hl7$$\"3O+]7`rg_[F*$\"3%R*fNXm%\\f\"F_o7$F]ilFg_l7&FailF\\il7$$\"3c*** \\(ovo$G&F*$\"3t!yA8Lkyg\"F_o7$FdilFg_l7&FhilFcil7$$\"3Y*\\(=#zMj_&F*$ \"3'e%)4@TKFh\"F_o7$F[jlFg_l7&F_jlFjil7$$\"3Y+]i:?)*odF*$\"3$*4g-9(zfh \"F_o7$FbjlFg_l7&FfjlFajl7$$\"3Y,D1R#H;,'F*$\"3*zSV(*p&p<;F_o7$FijlFg_ l7&F][mFhjl7$$\"3O,+]ikFaiF*$\"3d?bY2/(zh\"F_o7$F`[mFg_l7&Fd[mF_[m7$$ \"3D+++vNcTnF*$\"3!H'=v.,`9;F_o7$Fg[mFg_l7&F[\\mFf[m7$$\"3:****\\(o])G sF*$\"3l=K%\\)3N1;F_o7$F^\\mFg_l7&Fb\\mF]\\m7$$\"3?++]7@W)p(F*$\"3aQ2F w/i%f\"F_o7$Fe\\mFg_l7&Fi\\mFd\\m7$$\"3E,+]PN.o\")F*$\"3=^]j?\"*oz:F_o 7$F\\]mFg_l7&F`]mF[]m7$$\"3F**\\iS:!4-*F*$\"37@EH?N0Y:F_o7$Fc]mFg_l7&F g]mFb]m7$$\"3-++v3W].5F_o$\"3u\\p'ouW#)\\\"F_o7$Fj]mFg_l7&F^^mFi]m7$$ \"3-+++&e:%*3\"F_o$\"3-^PKOwi`9F_o7$Fa^mFg_l7&Fe^mF`^m7$$\"3'**\\ilq]$ *=\"F_o$\"31yz_()fI*R\"F_o7$Fh^mFg_l7&F\\_mFg^m7$$\"3))****\\A-\"yF\"F _o$\"3cnA\"Rm40N\"F_o7$F__mFg_l7&Fc_mF^_m7$$\"3?+DcJV'[P\"F_o$\"3]AQmq 3P(H\"F_o7$Ff_mFg_l7&Fj_mFe_m7$$\"3C+vo%z#Gn9F_o$\"3cv;!>77zC\"F_o7$F] `mFg_l7&Fa`mF\\`m7$$\"3O+]il\"F_o7$Fd`mFg_l7&Fh`m Fc`m7$$\"3/+D\"GmjAl\"F_o$\"3%=NLIWOS:\"F_o7$F[amFg_l7&F_amFj`m7$$\"3u **\\(o%)yxu\"F_o$\"3%[p9_7>(36F_o7$FbamFg_l7&FfamFaam7$$\"3'**\\i!zA*p %=F_o$\"3oqm\"R]iS1\"F_o7$FiamFg_l7&F]bmFham7$$\"3;+vVjyNL>F_o$\"3dZ$) [$H%>F5F_o7$F`bmFg_l7&FdbmF_bm7$$\"3()**\\ig]jE?F_o$\"3qrB=\"*zI%*)*F* 7$FgbmFg_l7&F[cmFfbm7$$\"3t****\\K&**H7#F_o$\"37$y$GSj$e_*F*7$F^cmFg_l 7&FbcmF]cm7$$\"3()**\\7oLF$[nHF _o$\"3Q$)*=qy(>sqF*7$F]gmFg_l7&FagmF\\gm7$$\"3/++vVK/gIF_o$\"3Qn%\\ob2 ,(oF*7$FdgmFg_l7&FhgmFcgm7$$\"3!)*\\i!R]%p:$F_o$\"3S%>6]YA%pmF*7$F[hmF g_l7&F_hmFjgm7$$\"3]+++&)HF]KF_o$\"35b#3rGjf['F*7$FbhmFg_l7&FfhmFahm7$ $\"3/+]P*G9dM$F_o$\"3F#=k.1`wI'F*7$FihmFg_l7&F]imFhhm7$$\"3E+Dc\"Hl.W$ F_o$\"30er*Q1*\\RhF*7$F`imFg_l7&FdimF_im7$$\"3x****\\K(Rt_$F_o$\"3AFKO ,u6#*fF*7$FgimFg_l7&F[jmFfim7$$\"3p**\\(oDAqi$F_o$\"3*>Caj'o1JeF*7$F^j mFg_l7&FbjmF]jm7$$\"3W+++&\\zhr$F_o$\"3c\"Ryyf'p$p&F*7$FejmFg_l7&FijmF djm7$$\"3m*\\ilqR7\"QF_o$\"3ia:z " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " The area of the region is given by " }{XPPEDIT 18 0 "Int((2*x+1)/(1+x^ 2),x=-1/2..4)" "6#-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF*F*F*F*F*,&F*F**$F+F )F*!\"\"/F+;,$*&F*F*F)F.F.\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "(2*x+1)/(1+x^2) = 2*x/(1+x^2) +1/(1+x^2)" "6#/*&,&*&\"\"#\"\"\"%\"xGF(F(F(F(F(,&F(F(*$F)F'F(!\"\",&* (F'F(F)F(,&F(F(*$F)F'F(F,F(*&F(F(,&F(F(*$F)F'F(F,F(" }{TEXT -1 7 ", so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+1)/(1 +x^2),x)=Int(2*x/(1+x^2),x)+Int(1/(1+x^2),x)" "6#/-%$IntG6$*&,&*&\"\"# \"\"\"%\"xGF+F+F+F+F+,&F+F+*$F,F*F+!\"\"F,,&-F%6$*(F*F+F,F+,&F+F+*$F,F *F+F/F,F+-F%6$*&F+F+,&F+F+*$F,F*F+F/F,F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(1+x^2)+arctan*x+c;" "6# /%!G,(-%#lnG6#,&\"\"\"F**$%\"xG\"\"#F*F**&%'arctanGF*F,F*F*%\"cGF*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The required area is " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int((2*x+1)/(1+x^2),x = -1/2 .. 4) = ln(1+x^2)+arcta n*x;" "6#/-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF+F+F+F+F+,&F+F+*$F,F*F+!\"\" /F,;,$*&F+F+F*F/F/\"\"%,&-%#lnG6#,&F+F+*$F,F*F+F+*&%'arctanGF+F,F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4,``],[-1/2,``])" "6#-%*PIE CEWISEG6$7$\"\"%%!G7$,$*&\"\"\"F,\"\"#!\"\"F.F(" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ln(17)+arctan(4)-( ln(5/4)+arctan(-1/2))" "6#/%!G,(-%#lnG6#\"#<\"\"\"-%'arctanG6#\"\"%F*, &-F'6#*&\"\"&F*F.!\"\"F*-F,6#,$*&F*F*\"\"#F4F4F*F4" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=ln(68/5)+arctan(4)+arctan(1/2)" "6#/%!G,(-%#lnG6#*& \"#o\"\"\"\"\"&!\"\"F+-%'arctanG6#\"\"%F+-F/6#*&F+F+\"\"#F-F+" }{TEXT -1 1 " " }{TEXT 303 1 "~" }{TEXT -1 14 " 4.399535066. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 66 "Int((2*x+1)/(1+x^2),x=-1/2..4);\n``=value(%) ;\ncombine(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ *&,&*&\"\"#\"\"\"%\"xGF*F*F*F*F*,&F*F**$)F+F)F*F*!\"\"/F+;#F/F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,-%#lnG6#\"#<\"\"\"-%'arctanG6# \"\"%F*-F'6#\"\"&!\"\"*&\"\"#F*-F'6#F4F*F*-F,6##F*F4F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(-%'arctanG6##\"\"*\"\"#!\"\"%#PiG\"\"\"-%#ln G6##\"#o\"\"&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+m]`*R%!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "The area under " }{XPPEDIT 18 0 "y = sqrt(a^2-x^2);" "6#/%\"yG-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"xGF+! \"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 41 "In this section we consider the integral " }{XPPEDIT 18 0 "Int(sqrt(a^2-x^2),x);" "6#-%$IntG6$-%%sqrtG6#,&*$%\"aG\"\"#\"\" \"*$%\"xGF,!\"\"F/" }{TEXT -1 9 " , where " }{TEXT 284 1 "a" }{TEXT -1 25 " is a positive constant. 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B%[F1$\"3y4,7\")p2pjF17$$\"3a@Y8*Hq['[F1$\"3E%eS$4EM`jF17$$\"3'G'*4tDG x)[F1$\"3rqP8Fs\"yL'F17$$\"3YY9TNI*3jF17$$\"3)\\vl1KuX&\\F1$\"33r;4hB)\\H'F17$$\"3Kzb*4 QZz(\\F1$\"31ANLe)p3G'F17$$\"3]4C][]V,]F1$\"3s2G1f97niF17$$\"3`J54zyyD ]F1$\"3q\"f1ZN9LD'F17$$\"3ma\"4M7'RZ]F1$\"3yYL,BkVTiF17$$\"3+n8Cz\\*=2 &F1$\"3-%fVbh%QGiF17$$\"33()>A7Vn'4&F1$\"3b4L2,Ei:iF17$$\"3$yM*>])>27& F1$\"3X\\$o<#>l.iF17$$\"3plI.@IpU^F1$\"3+#pOJ0iI>'F17$$\"3E3;`Vp)*o^F1 $\"3i67j2'>3='F17$$\"3UpQp/$)R\">&F1$\"3OV\\^&)yuqhF17$$\"3`\"G?7O@w@& F1$\"3GD7JV&z$fhF17$$\"3a5NW#pm4C&F1$\"3!fw[))=I'\\hF17$$\"3+6cm5)>nE& F1$\"3_rZ0\"Hu#RhF17$$\"33Pu&pAu8H&F1$\"3YJ>aOhuHhF17$$\"32!=shRIsJ&F1 $\"3(>_aSv_,7'F17$$\"3dU57**))3T`F1$\"3r*e:t#yl6hF17$$\"3gCE%*[=%pO&F1 $\"3,H'\\KQMG5'F17$$\"3)yjf4y?RR&F1$\"3K`\\/xS/%4'F17$$\"3cUdsJb]1u#QY1'F17$$\"3xySQA0$4_&F1$ \"3'*4Ii6WAegF17$$\"3wXbJAT(yXgF17$$\"3jd!=V\"**o,cF1$\"3`(fgB,k+/'F17$$\"36vq+rwREcF1$ \"3dWy^9X?NgF17$$\"3[V&[ESxMl&F1$\"3O7cE;#[-.'F17$$\"3k0c$ROZ**o9x0dF1$\"3ir4J#*4w@gF17$$\"3]0n*pm!)>t&F1$\" 3yM4n]*R!=gF17$$\"3idfj,&p%fdF1$\"31qIr6k^9gF17$$\"3)*GXER#))fy&F1$\"3 DgZaCK[6gF17$$\"3\"fo,T'o98eF1$\"3-:7'RlZ(3gF17$$\"34GBP*)\\6SeF1$\"32 `\"3v/,k+'F17$$\"3)4SAp$=#\\'eF1$\"37omJTnc/gF17$$\"3whQ*3vyL*eF1$\"3F )*z$[/WG+'F17$$\"3_B4Z![\\)=fF1$\"34==))Hqk,gF17$$\"331+w>5-YfF1$\"3'* y(y_cG2+'F17$$\"3U\"3r#fq.sfF1$\"3ITx1\"\\&>+gF17$$\"3#)>1B,+++gF1$\"3 )3+++++++'F1F[\\l-F$6$7%7$$\"3U+++++++bF1F`[l7$Fg[n$\"3G+++++++]Fd_l7$ Ff[lFj[nF[\\l-%%TEXTG6%7$$\"\"$F)$\"\"&F)Q\"a6\"-%%FONTG6$%*HELVETICAG \"\"*-F^\\n6%7$Fa\\n$!\"&!\"#Q\"xFf\\nFg\\n-F^\\n6%7$F_]n$\"#7F)Q\"yFf \\nFg\\n-F^\\n6%7$F_]n$\"#&*Fa]nQ\"AFf\\nFg\\n-F^\\n6%7$$\"#jFa]n$\"#( )Fa]nQ\"BFf\\nFg\\n-F^\\n6%7$Fb^nF_]nQ\"CFf\\nFg\\n-F^\\n6%7$F_]nF_]nQ \"OFf\\nFg\\n-F^\\n6%7$$Fd\\nFa]n$\"#:Fa]nQ\"qFf\\n-Fh\\n6$%'SYMBOLGF_ [l-F^\\n6%7$$\"#bFa]n$\"#nFa]nFe_nFf_n-%+AXESLABELSG6%%!GFc`n-Fh\\n6#% (DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%%VIEWG6$;$F)F )Ff]nFaan" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14 " }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "One way to find the \+ indefinite integral " }{XPPEDIT 18 0 "Int(sqrt(a^2-x^2),x)" "6#-%$IntG 6$-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"xGF,!\"\"F/" }{TEXT -1 132 " is t o find a formula for the area of the region OABC in the picture as the sum of the areas of the sector OAB and the triangle OBC." }}{PARA 0 " " 0 "" {TEXT -1 30 "The area of the sector OAB is " }{XPPEDIT 18 0 "1/ 2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*theta; " "6#*&%\"aG\"\"#%&thetaG\"\"\"" }{TEXT -1 32 ", where angle AOB = ang le OBC = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 33 "The area of the triangle OBC is " } {XPPEDIT 18 0 "x*sqrt(a^2-x^2)/2;" "6#*(%\"xG\"\"\"-%%sqrtG6#,&*$%\"aG \"\"#F%*$F$F,!\"\"F%F,F." }{TEXT -1 13 ", since BC = " }{XPPEDIT 18 0 "sqrt(a^2-x^2);" "6#-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"xGF)!\"\"" } {TEXT -1 24 " by Pythagoras' theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 61 "This shows that the area of the regio n OABC under the graph " }{XPPEDIT 18 0 "y = sqrt(a^2-x^2);" "6#/%\"y G-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"xGF+!\"\"" }{TEXT -1 3 " is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "a^2*theta/2+x*sqrt(a^2-x^2)/2 = a^2/2;" "6#/,&*(%\"aG\" \"#%&thetaG\"\"\"F'!\"\"F)*(%\"xGF)-%%sqrtG6#,&*$F&F'F)*$F,F'F*F)F'F*F )*&F&F'F'F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x/a)+x*sqrt(a^2-x^ 2)/2" "6#,&-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"F)*(F(F)-%%sqrtG6#,&*$F* \"\"#F)*$F(F2F+F)F2F+F)" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 20 "using the fact that " }{XPPEDIT 18 0 "sin*theta = x/a;" "6#/*&% $sinG\"\"\"%&thetaGF&*&%\"xGF&%\"aG!\"\"" }{TEXT -1 11 ", that is, " } {XPPEDIT 18 0 "theta=arcsin(x/a)" "6#/%&thetaG-%'arcsinG6#*&%\"xG\"\" \"%\"aG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follo ws that " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sqr t(a^2-x^2),x) = a^2/2;" "6#/-%$IntG6$-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$% \"xGF-!\"\"F0*&F,F-F-F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x/a)+x *sqrt(a^2-x^2)/2+c" "6#,(-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"F)*(F(F)-% %sqrtG6#,&*$F*\"\"#F)*$F(F2F+F)F2F+F)%\"cGF)" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 304 27 "______________________ _____" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The integral can also be found by means of the inverse substitution " }{XPPEDIT 18 0 "x = a*sin*theta;" "6#/%\"xG*(%\"aG\"\" \"%$sinGF'%&thetaGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "-Pi/2<=th eta" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/2 " "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 12 ", that is, " } {XPPEDIT 18 0 "theta=arcsin(x/a)" "6#/%&thetaG-%'arcsinG6#*&%\"xG\"\" \"%\"aG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(a^2-x^2),x) " "6#-%$IntG6$-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"xGF,!\"\"F/" }{TEXT -1 11 " ... " }{XPPEDIT 18 0 "PIECEWISE([x = a*sin*theta, theta \+ = arcsin(x/a)],[dx = a*cos*theta*`.`*d*theta, ``]);" "6#-%*PIECEWISEG6 $7$/%\"xG*(%\"aG\"\"\"%$sinGF+%&thetaGF+/F--%'arcsinG6#*&F(F+F*!\"\"7$ /%#dxG*.F*F+%$cosGF+F-F+%\".GF+%\"dGF+F-F+%!G" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(``(sqrt(a^2-a^2*sin^2*theta))*a*cos*theta,thet a);" "6#/%!G-%$IntG6$**-F$6#-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*(F0F1%$sinG F1%&thetaGF2!\"\"F2F0F2%$cosGF2F5F2F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(``(a*sqrt(1-sin^2*theta)) *a*cos*theta,theta);" "6#/%!G-%$IntG6$**-F$6#*&%\"aG\"\"\"-%%sqrtG6#,& F-F-*&%$sinG\"\"#%&thetaGF-!\"\"F-F-F,F-%$cosGF-F5F-F5" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(a^2*cos^2* theta,theta)" "6#/%!G-%$IntG6$*(%\"aG\"\"#%$cosGF*%&thetaG\"\"\"F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "( using the relation " } {XPPEDIT 18 0 "cos*theta = ``;" "6#/*&%$cosG\"\"\"%&thetaGF&%!G" } {TEXT 308 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1-sin^2*theta)" " 6#-%%sqrtG6#,&\"\"\"F'*&%$sinG\"\"#%&thetaGF'!\"\"" }{TEXT -1 36 ", wh ere the + sign is taken because " }{XPPEDIT 18 0 "-Pi/2<=theta" "6#1,$ *&%#PiG\"\"\"\"\"#!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/2" "6#1%!G*& %#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a^2*Int(1/2+cos*2*theta/2,theta);" "6#/%!G* &%\"aG\"\"#-%$IntG6$,&*&\"\"\"F-F'!\"\"F-**%$cosGF-F'F-%&thetaGF-F'F.F -F1F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "( using the form ula " }{XPPEDIT 18 0 "cos^2*theta = 1/2+cos*2*theta/2;" "6#/*&%$cosG\" \"#%&thetaG\"\"\",&*&F(F(F&!\"\"F(**F%F(F&F(F'F(F&F+F(" }{TEXT -1 4 " \+ ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a^2*(thet a/2+sin*2*theta/4)+c;" "6#/%!G,&*&%\"aG\"\"#,&*&%&thetaG\"\"\"F(!\"\"F ,**%$sinGF,F(F,F+F,\"\"%F-F,F,F,%\"cGF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = a^2*(theta/2+sin*theta*cos*theta/2)+c;" "6#/%!G,&*&%\"aG\"\"#,&*& %&thetaG\"\"\"F(!\"\"F,*,%$sinGF,F+F,%$cosGF,F+F,F(F-F,F,F,%\"cGF," } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=a ^2/2" "6#/%!G*&%\"aG\"\"#F'!\"\"" }{XPPEDIT 18 0 "``(theta+sin*theta*c os*theta)+c;" "6#,&-%!G6#,&%&thetaG\"\"\"**%$sinGF)F(F)%$cosGF)F(F)F)F )%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "sin*theta = x/a;" "6#/*&%$s inG\"\"\"%&thetaGF&*&%\"xGF&%\"aG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-Pi/2<=theta" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F)%&thetaG" } {XPPEDIT 18 0 "``<=Pi/2" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 19 ", it follows that " }{XPPEDIT 18 0 "cos*theta = sqrt(a^2-x^2)/a;" "6 #/*&%$cosG\"\"\"%&thetaGF&*&-%%sqrtG6#,&*$%\"aG\"\"#F&*$%\"xGF/!\"\"F& F.F2" }{TEXT -1 87 ", as suggested by applying Pythagoras' theorem in \+ the following right-angled triangle. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 212 128 128 {PLOTDATA 2 "6+-%'CURVESG6$7&7$$\"\"!F)F(7$ $\"\"#F)F(7$F+$\"\"\"F)F'-%&COLORG6&%$RGBG$\"#X!\"#F)$\"#&)F6-F$6$7S7$ $\"3')*************4'!#=F(7$$\"3E*)*\\,\\)o*4'F?$\"3+.\"\\=4wY;'!#?7$$ \"35RH3%f5*)4'F?$\"3UnFnsP!G:\"!#>7$$\"3;\"**fpQsu4'F?$\"3cRX^M8'ev\"F K7$$\"3O%fy)=CU&4'F?$\"35VN!4!GuiBFK7$$\"3g]UqyCy#4'F?$\"3wNCE&\\2l'HF K7$$\"3o-_vj1!)*3'F?$\"3i(e'Q@e,ENFK7$$\"3s4Z,S<<'3'F?$\"3fL=V2-/0TFK7 $$\"3k4B@6'R=3'F?$\"3%fXBi$oZ.ZFK7$$\"3c3Ae!zNp2'F?$\"3-`lii9a*H&FK7$$ \"3Ex%y.7#GrgF?$\"3#p:'4.^87fFK7$$\"3eZkR^3zlgF?$\"3=2=Hz=@^kFK7$$\"3u Ep0%GO!fgF?$\"3eK'**RStu0(FK7$$\"3QrQ!pJW;0'F?$\"3gP?x$39bm(FK7$$\"3]X z*HWVR/'F?$\"3;FEPZLu]#)FK7$$\"3)GP=A)*fk.'F?$\"3ia^w.(>:y)FK7$$\"36M& Q>\"[&p-'F?$\"3t^r#=ByOG\")=^(fF?$\"3Y(zeBE))zA\"F?7$ $\"3U![e5iBE'fF?$\"3;\"oQtb\"H(G\"F?7$$\"3O!*[Z'oI1&fF?$\"3K1#o=rQ;M\" F?7$$\"3sG;?9%Rr$fF?$\"3JB(zz7M,S\"F?7$$\"3N:K$Q'p^AfF?$\"3Wd,Oq2vg9F? 7$$\"3(p4&*==%G4fF?$\"3Ft1!*)o$R8:F?7$$\"3W+txnoY%*eF?$\"3')opDbZ6q:F? 7$$\"3Q>R#)*Q(eyeF?$\"3ZB(=k!3cG;F?7$$\"3c;F?7 $$\"3qLW\"fy(GudF?$\"3w_i;7;im>F?7$$\"3W!y/2u*\\adF?$\"3^`ajs;zB?F?7$$ \"3\\L1]`QONdF?$\"3K:%[sd-u2#F?7$$\"3EIj+zu#[r&F?$\"3jDgCk)\\3l(=#F?7$$\"3qD*QXj)4scF?$\"36ae@jBRWAF?7$$\"3= I]E=]D]cF?$\"3cth(>vH))H#F?7$$\"3![jo)ekPFcF?$\"3![&ows*zUN#F?7$$\"3mU Z1?*\\Tg&F?$\"3!QiUl@Y!4CF?7$$\"3'>7t,?PBe&F?$\"3Qgq)*R)o\"fCF?7$$\"3w _*pi:'ycbF?$\"3K<'yZ\">P;DF?7$$\"3)e%*QVLOM`&F?$\"3w/8kI.JnDF?7$$\"3Oa (*)>FE!3bF?$\"33THnrDQ@EF?7$$\"3Q4&Hno5K[&F?$\"37>)e8:,Hn#F?7$$\"3=&o, ^'e+caF?$\"3KX`aKH+GFF?-%'COLOURG6&F3F)F)F)-F$6$7%7$$\"3!************* **=!# " 0 "" {MPLTEXT 1 0 195 "assume(a_>0):\nInt(sqrt(a^2-x^2),x);\nstudent[c hangevar](theta=arcsin(x/a),%,theta);\nsimplify(%);\ncombine(%,trig); \nvalue(%);\nexpand(%);\nsubs(theta=arcsin(x/a),%);\nsubs(a_=a,simplif y(subs(a=a_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&*$)% \"aG\"\"#\"\"\"F,*$)%\"xGF+F,!\"\"#F,F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*$)%\"aG\"\"#\"\"\"F+*&)-%$sinG6#%&thetaGF*F +F(F+!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"aG\"\"#\"\"\"-%$ IntG6$*$)-%$cosG6#%&thetaGF&F'F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&)%\"aG\"\"#\"\"\"-%$IntG6$,&*&#F'F&F'-%$cosG6#,$*&F&F'%&thetaGF'F'F' F'F-F'F3F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"aG\"\"#\"\"\",&*&# F'\"\"%F'-%$sinG6#,$*&F&F'%&thetaGF'F'F'F'*&F&!\"\"F1F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*()%\"aGF'F&-%$sinG6#%&the taGF&-%$cosGF-F&F&F&*(F'!\"\"F*F'F.F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*()%\"aGF'F&-%$sinG6#-%'arcsinG6#*&%\"xGF&F*! \"\"F&-%$cosGF-F&F&F&*&F%F&*&F)F&F.F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#!\"\"%\"xG\"\"\",&*$)%\"aGF%F(F(*$)F'F%F(F&#F( F%F(*&F/F(*&F+F(-%'arcsinG6#*&F'F(F,F&F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Maple gives a slightly di fferent result when the integral is calculated directly. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Int(s qrt(a^2-x^2),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*$,&*$)%\"aG\"\"#\"\"\"F,*$)%\"xGF+F,!\"\"#F,F+F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*(\"\"#!\"\"%\"xG\"\"\",&*$)%\"aGF%F(F(*$)F'F%F(F&# F(F%F(*&F/F(*&F+F(-%'arctanG6#*&F'F(F)#F&F%F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[f(x), `|`, `f '`(x)], [_________, ``, _ _________], [arcsin*x, `|`, 1/sqrt(1-x^2)], [arctan*x, `|`, 1/(1+x^2)] ]);" "6#-%'matrixG6#7&7%-%\"fG6#%\"xG%\"|grG-%$f~'G6#F+7%%*_________G% !G%+__________G7%*&%'arcsinG\"\"\"F+F7F,*&F7F7-%%sqrtG6#,&F7F7*$F+\"\" #!\"\"F?7%*&%'arctanGF7F+F7F,*&F7F7,&F7F7*$F+F>F7F?" }{TEXT -1 29 " \+ " }{XPPEDIT 18 0 "matrix([[f(x), `|`, Int(f( x),x)], [____________, ``, __________________], [1/sqrt(a^2-x^2), `|`, arcsin(x/a)+c], [1/(a^2+x^2), `|`, ``(1/a)*arctan(x/a)+c]]);" "6#-%'m atrixG6#7&7%-%\"fG6#%\"xG%\"|grG-%$IntG6$-F)6#F+F+7%%-____________G%!G %3__________________G7%*&\"\"\"F8-%%sqrtG6#,&*$%\"aG\"\"#F8*$F+F?!\"\" FAF,,&-%'arcsinG6#*&F+F8F>FAF8%\"cGF87%*&F8F8,&*$F>F?F8*$F+F?F8FAF,,&* &-F46#*&F8F8F>FAF8-%'arctanG6#*&F+F8F>FAF8F8FGF8" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 10 "Find (a) " }{XPPEDIT 18 0 "Int(1/(x^2+4),x);" "6#-%$IntG6$*&\"\"\"F',&*$%\"xG\"\"#F'\"\"%F'!\"\"F*" }{TEXT -1 13 " \+ and (b) " }{XPPEDIT 18 0 "Int(1/(x^2+4),x = 0 .. 2);" "6#-%$IntG6$* &\"\"\"F',&*$%\"xG\"\"#F'\"\"%F'!\"\"/F*;\"\"!F+" }{TEXT -1 2 ". " }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "arctan(x/2)+c" "6#,&-%'arctanG6#*&%\"xG\"\"\"\"\"#! \"\"F)%\"cGF)" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "Pi/8" "6#*&%#PiG \"\"\"\"\")!\"\"" }{TEXT -1 1 " " }{TEXT 311 1 "~" }{TEXT -1 9 " 0.392 70 " }}}{PARA 0 "" 0 "" {TEXT -1 40 "_________________________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "__ ______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 10 "Find (a) " }{XPPEDIT 18 0 "Int(1/sqrt(3- x^2),x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"$F'*$%\"xG\"\"#!\"\"F0 F." }{TEXT -1 13 " and (b) " }{XPPEDIT 18 0 "Int(1/sqrt(3-x^2),x = 0 .. 3/2);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"$F'*$%\"xG\"\"#!\" \"F0/F.;\"\"!*&F,F'F/F0" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "arcsin(x/sqrt(3))+c" "6#,&-%'arcsinG6#*&%\"xG\"\"\"-%%sqrtG6#\"\"$! \"\"F)%\"cGF)" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "Pi/3" "6#*&%#PiG \"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 310 1 "~" }{TEXT -1 8 " 1.047 2 " }}}{PARA 0 "" 0 "" {TEXT -1 40 "__________________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "__ ______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 10 "Find (a) " }{XPPEDIT 18 0 "Int(1/sqrt(4- 3*x^2),x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"%F'*&\"\"$F'*$%\"xG \"\"#F'!\"\"F2F0" }{TEXT -1 13 " and (b) " }{XPPEDIT 18 0 "Int(1/s qrt(4-3*x^2),x = -1 .. 1);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"%F'* &\"\"$F'*$%\"xG\"\"#F'!\"\"F2/F0;,$F'F2F'" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) \+ " }{XPPEDIT 18 0 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(sqrt(3)/2*x)+c" "6#,&-%'arcsinG6 #*(-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"%\"xGF,F,%\"cGF," }{TEXT -1 7 " (b ) " }{XPPEDIT 18 0 "2*Pi/(3*sqrt(3))=2*sqrt(3)*Pi/9" "6#/*(\"\"#\"\"\" %#PiGF&*&\"\"$F&-%%sqrtG6#F)F&!\"\"**F%F&-F+6#F)F&F'F&\"\"*F-" }{TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________________ ____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 " Find \+ (a) " }{XPPEDIT 18 0 "Int(1/(x^2+2),x);" "6#-%$IntG6$*&\"\"\"F',&*$% \"xG\"\"#F'F+F'!\"\"F*" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "Int(1/( x^2+2*x+3),x)" "6#-%$IntG6$*&\"\"\"F',(*$%\"xG\"\"#F'*&F+F'F*F'F'\"\"$ F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) \+ " }{XPPEDIT 18 0 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/sqrt(2))+c" "6#,&-%'arctanG6#* &%\"xG\"\"\"-%%sqrtG6#\"\"#!\"\"F)%\"cGF)" }{TEXT -1 11 " (b) " }{XPPEDIT 18 0 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan((x+1)/sqrt(2))+c;" "6#,&-%'arctanG6#*& ,&%\"xG\"\"\"F*F*F*-%%sqrtG6#\"\"#!\"\"F*%\"cGF*" }{TEXT -1 2 " " }}} {PARA 0 "" 0 "" {TEXT -1 40 "________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________ ____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(1/(x^2+6*x+13),x)" "6#-%$IntG6 $*&\"\"\"F',(*$%\"xG\"\"#F'*&\"\"'F'F*F'F'\"#8F'!\"\"F*" }{TEXT -1 2 " . " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan((x+3)/2)+c" "6#,&-%'arctanG6#*&,&%\"xG \"\"\"\"\"$F*F*\"\"#!\"\"F*%\"cGF*" }{TEXT -1 2 " " }}}{PARA 0 "" 0 " " {TEXT -1 40 "________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(1/sqrt(8-2*x-x^2),x)" "6#-%$IntG6$*&\" \"\"F'-%%sqrtG6#,(\"\")F'*&\"\"#F'%\"xGF'!\"\"*$F/F.F0F0F/" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin((x+1)/3)+c" "6#,&-%'arcsinG6#*&, &%\"xG\"\"\"F*F*F*\"\"$!\"\"F*%\"cGF*" }{TEXT -1 3 " " }}}{PARA 0 " " 0 "" {TEXT -1 40 "________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 54 " Find the area of the region bounded by the graph of " }{XPPEDIT 18 0 "y=x^2/(1+x^2)" "6#/%\"yG*&%\"xG\"\"#,&\"\"\"F)*$F&F'F)!\"\"" } {TEXT -1 6 ", the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 19 " axis a nd the line " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 2 ". " }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^2/(1+x^2),x = 0 .. 4) = 4-arctan*4;" "6#/ -%$IntG6$*&%\"xG\"\"#,&\"\"\"F+*$F(F)F+!\"\"/F(;\"\"!\"\"%,&F1F+*&%'ar ctanGF+F1F+F-" }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 14 " 2.67418 2336. " }}}{PARA 0 "" 0 "" {TEXT -1 40 "______________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "__ ______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q8" }} {PARA 0 "" 0 "" {TEXT -1 94 "Use two methods to find the area of the r egion in the first quadrant bounded by the graph of " }{XPPEDIT 18 0 "y = sqrt(4-x^2);" "6#/%\"yG-%%sqrtG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 6 ", the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " }{TEXT 312 1 "y" }{TEXT -1 19 " axes and the line " }{XPPEDIT 18 0 "x= 1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 148 "( a) Use some trigonometry in connection with the following picture in w hich the curve is an arc of a circle with centre at the origin and rad ius 2. 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" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (sqrt(4-x^2),x = 0 .. 1)=Pi/3+sqrt(3)/2" "6#/-%$IntG6$-%%sqrtG6#,&\"\" %\"\"\"*$%\"xG\"\"#!\"\"/F.;\"\"!F,,&*&%#PiGF,\"\"$F0F,*&-F(6#F7F,F/F0 F," }{TEXT -1 1 " " }{TEXT 313 1 "~" }{TEXT -1 15 " 1.913222955. " }} }{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 40 "________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q9" }}{PARA 257 "" 0 "" {TEXT -1 10 "Show that " } {XPPEDIT 18 0 "Int(x^4*(1-x)^4/(1+x^2),x = 0 .. 1)=22/7-Pi" "6#/-%$Int G6$*(%\"xG\"\"%,&\"\"\"F+F(!\"\"F),&F+F+*$F(\"\"#F+F,/F(;\"\"!F+,&*&\" #AF+\"\"(F,F+%#PiGF," }{TEXT -1 3 ". " }}{PARA 257 "" 0 "" {TEXT -1 44 "Hint: Use polynomial division to show that " }{XPPEDIT 18 0 "x^4* (1-x)^4/(1+x^2)=x^6-4*x^5+5*x^4-4*x^2+4-4/(1+x^2)" "6#/*(%\"xG\"\"%,& \"\"\"F(F%!\"\"F&,&F(F(*$F%\"\"#F(F),.*$F%\"\"'F(*&F&F(*$F%\"\"&F(F)*& F2F(*$F%F&F(F(*&F&F(*$F%F,F(F)F&F(*&F&F(,&F(F(*$F%F,F(F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 131 "Remark: This integral was known by K. Mahler in the mid-1960s and appears in an exam at the Universit y of Sydney in November 1960. " }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ See: " }{URLLINK 17 "http://mathworld.wolfram.com/PiFormulas.ht ml" 4 "http://mathworld.wolfram.com/PiFormulas.html" "" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 40 "_____________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "Arcsin graphs " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 619 "p1 := plot(sin(Pi*x),x=-1..-1/2,color=red):\np2 := plot(sin(Pi*x) ,x=1/2..3/2,color=red):\np3 := plot(sin(Pi*x),x=-1/2..1/2,color=blue,t hickness=3):\np4 := plot([[[1/2,1],[-1/2,-1]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=blue):\nt1 := plots[text plot]([[1.7,-.1,`x`],[-.05,1.3,`y`]],\n color=COLOR (RGB,.01,.01,.01),font=[HELVETICA,9]):\nplots[display]([p1,p2,p3,p4,t1 ],ytickmarks=3,\nxtickmarks=[-1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1 .5=`3p/2`],\n font=[SYMBOL,9],labels=[``,``],label_font=[HELVETICA,9 ],title=\"y = sin x\",\n title_font=[HELVETICA,9],view=[-1..1.7,-1.. 1.3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 698 "p1 := plot(sin(Pi*x)/Pi,x=-1/2..1/2,color=blue):\np2 := plot(arcsin(Pi*x)/Pi,x=-1/Pi..1/Pi,color=red,numpoints=100):\np3 : = plot(x,x=-1/2..1/2,color=green):\nt1 := plots[textplot]([.5,.27,`y = sin x`],\n color=blue,font=[HELVETICA,9]):\n t2 := plots[textplot]([.2,.55,`y = arcsin x`],\n \+ color=red,font=[HELVETICA,9]):\nt3 := plots[textplot]([[.6,-.03,`x `],[-.03,.6,`y`]],\n color=black,font=[HELVET ICA,9]):\nplots[display]([p1,p2,p3,t1,t2,t3],labels=[``,``],\nxtickmar ks=[-.5=`-p/2`,-.31831=`-1`,0=`0`,.31831=`1`,.5=`p/2`],\nytickmarks=[- .5=`-p/2`,-.31831=`-1`,0=`0`,.31831=`1`,.5=`p/2`],\n font=[SYMBOL,9] ,view=[-.5..0.6,-0.5..0.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "p1 := plot(arcsin(Pi*x)/Pi, x=-1/Pi..1/Pi,color=red,numpoints=100):\n\nt1 := plots[textplot]([.2,. 48,`y = arcsin x`],\n color=red,font=[HELVETI CA,9]):\nt2 := plots[textplot]([[.37,-.03,`x`],[-.02,.6,`y`]],\n \+ color=black,font=[HELVETICA,9]):\nplots[display]([ p1,t1,t2],labels=[``,``],\nxtickmarks=[-.31831=`-1`,0=`0`,.31831=`1`], \nytickmarks=[-.5=`-p/2`,0=`0`,.5=`p/2`],font=[SYMBOL,9],\n view=[-.3 3..0.37,-0.5..0.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 497 "p1 := plot([arcsin(x),1/sqrt(1-x^2)],x=- 1..1,y=-Pi/2..3,numpoints=100,\n color=[red,blue]):\np2 := plot([ [[-1,-Pi/2],[-1,3]],[[1,-Pi/2],[1,3]]],\n color=black,linesty le=3):\nt1 := plots[textplot]([-.6,-1.4,`y = arcsin x`],color=red):\nt 2 := plots[textplot]([[1.2,-.1,`x`],[-.1,3,`y`]],\n \+ color=black):\nplots[display]([p1,p2,t1,t2],labels=[``,``],\n \+ xtickmarks=[-1=`-1`,-.5=`-.5`,.5=`.5`,0=`0`,1=`1`],\n ytickmarks=[- 1=`-1`,1=`1`,2=`2`],view=[-1.2..1.2,-1.6..3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "f := x -> 1 /sqrt(1-x^2):\np1 := plot(f(x),x=-1..1,y=0..3,color=red):\np2 := plot( [[[-1,-Pi/2],[-1,3]],[[1,-Pi/2],[1,3]]],\n color=black,linest yle=3):\na := -1/2: b := 1/2:\np3 := plot(f(x),x=a..b,color=COLOR(RGB, .87,.87,.93),filled=true):\np4 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b )]]],color=black):\nt1 := plots[textplot]([[1.1,-.1,`x`],[-.1,3,`y`]], \n color=black):\nplots[display]([p1,p2,p3,p4 ,t1],labels=[``,``],\nxtickmarks=[-1=`-1`,-.5=`-.5`,.5=`.5`,0=`0`,1=`1 `],\n ytickmarks=[-1=`-1`,1=`1`,2=`2`],\n view=[-1.1..1.1,-.1..3]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "Arccos gra phs " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "p1 := plot(cos(Pi*x),x=-1/2..0,color=red):\np2 := pl ot(cos(Pi*x),x=1..2,color=red):\np3 := plot(cos(Pi*x),x=0..1,color=blu e,thickness=3):\np4 := plot([[[0,1],[1,-1]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=blue):\nt1 := plots[textpl ot]([[2.2,-.1,`x`],[-.05,1.3,`y`]],\n color=COLOR(R GB,.01,.01,.01),font=[HELVETICA,9]):\nplots[display]([p1,p2,p3,p4,t1], ytickmarks=3,\nxtickmarks=[-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/2`, 2=`2p`],\n font=[SYMBOL,9],labels=[``,``],label_font=[HELVETICA,9],t itle=\"y = cos x\",\n title_font=[HELVETICA,9],view=[-.5..2.2,-1..1. 3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "p1 := plot(cos(Pi*x)/Pi,x=-0..1,color=blue):\np2 := \+ plot(arccos(Pi*x)/Pi,x=-1/Pi..1/Pi,color=red,numpoints=100):\np3 := pl ot(x,x=-.2..0.7,color=green):\nt1 := plots[textplot]([.5,.27,`y = cos \+ x`],\n color=blue,font=[HELVETICA,9]):\nt2 := plots[textplot]([.2,.55,`y = arccos x`],\n c olor=red,font=[HELVETICA,9]):\nt3 := plots[textplot]([[1.13,-.03,`x`], [-.03,1.13,`y`]],\n color=black,font=[HELVETI CA,9]):\nplots[display]([p1,p2,p3,t1,t2,t3],labels=[``,``],\nxtickmark s=[-.31831=`-1`,0=`0`,.31831=`1`,.5=`p/2`,.63662=`2`,1=`p`],\nytickmar ks=[-.31831=`-1`,0=`0`,.31831=`1`,.5=`p/2`,.63662=`2`,1=`p`],\n font =[SYMBOL,9],view=[-.33..1.13,-.33..1.13]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "p1 := plot(arccos (Pi*x)/Pi,x=-1/Pi..1/Pi,color=red,numpoints=100):\nt1 := plots[textplo t]([-.2,.96,`y = arccos x`],\n color=red,font =[HELVETICA,9]):\nt2 := plots[textplot]([[.37,-.03,`x`],[-.02,1.13,`y` ]],\n color=black,font=[HELVETICA,9]):\nplots [display]([p1,t1,t2],labels=[``,``],\nxtickmarks=[-.31831=`-1`,0=`0`,. 31831=`1`],\nytickmarks=[0=`0`,.5=`p/2`,1=`p`],\n font=[SYMBOL,9],vi ew=[-.33..0.37,-.03..1.13]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 15 "Arctan graphs " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "p1 := plot(tan(Pi*x),x=-3/2 ..-1/2,y=-6..6,color=red,discont=true):\np2 := plot(tan(Pi*x),x=1/2..3 /2,color=red,discont=true):\np3 := plot(tan(Pi*x),x=-1/2..1/2,color=bl ue,thickness=2):\np4 := plot([[[-1.5,-6],[-1.5,6]],[[-.5,-6],[-.5,6]], \n [[.5,-6],[.5,6]],[[1.5,-6],[1.5,6]]],color=black,linestyle=3 ):\nt1 := plots[textplot]([[1.7,-.2,`x`],[-.06,6,`y`]],\n \+ color=COLOR(RGB,.01,.01,.01),font=[HELVETICA,9]):\nplots[displa y]([p1,p2,p3,p4,t1],ytickmarks=3,\nxtickmarks=[-1.5=`-3p/2`,-1=`-p`,-. 5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/2`],\n font=[SYMBOL,9],labels= [``,``],label_font=[HELVETICA,9],title=\"y = tan x\",\n title_font=[ HELVETICA,9],view=[-1.6..1.7,-6..6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1182 "p1 := plot(tan(Pi*x)/P i,x=-1/2..1/2,y=-2.1..2.1,color=blue,discont=true):\np2 := plot(arctan (Pi*x)/Pi,x=-2.1..2.1,color=red):\np3 := plot(x,x=-2..2,color=green): \np4 := plot([[[-.5,-2.1],[-.5,2.1]],[[.5,-2.1],[.5,2.1]]],\n \+ color=COLOR(RGB,.3,0,.7),linestyle=3):\np5 := plot([[[-2.1,-.5],[2.1, -.5]],[[-2.1,.5],[2.1,.5]]],\n color=brown,linestyle=3):\nt1 \+ := plots[textplot]([.8,1.8,`y = tan x`],\n co lor=blue,font=[HELVETICA,9]):\nt2 := plots[textplot]([1.8,.35,`y = arc tan x`],\n color=red,font=[HELVETICA,9]):\nt3 := plots[textplot]([[2.1,-.07,`x`],[-.07,2.1,`y`]],\n \+ color=black,font=[HELVETICA,9]):\nt4 := plots[textplot]([[-. 4,-.07,`-p/2`],[.4,-.07,`p/2`],\n [-.1,-.4,`-p/2`],[-.1,.4 ,`p/2`]],\n color=black,font=[SYMBOL,9]):\npl ots[display]([p1,p2,p3,p4,p5,t1,t2,t3,t4],labels=[``,``],\nxtickmarks= [-1.9099=`-6`,-1.2732=`-4`,-.63662=`-2`,0=`0`,\n \+ .63662=`2`,1.2732=`4`,1.9099=`6`],\nytickmarks=[-1.9099=`-6`,-1.2732= `-4`,-.63662=`-2`,0=`0`,\n .63662=`2`,1.2732=`4` ,1.9099=`6`],\n font=[SYMBOL,9],view=[-2.1..2.1,-2.1..2.1]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 702 "p1 := plot(arctan(Pi*x)/Pi,x=-2.1..2.1,color=red):\np2 := plot([[ [-2.1,-.5],[2.1,-.5]],[[-2.1,.5],[2.1,.5]]],\n color=black,li nestyle=3):\nt1 := plots[textplot]([1.8,.4,`y = arctan x`],\n \+ color=red,font=[HELVETICA,9]):\nt2 := plots[textplot] ([[2.1,-.07,`x`],[-.07,.6,`y`]],\n color=blac k,font=[HELVETICA,9]):\nt3 := plots[textplot]([[-.13,-.44,`-p/2`],[-.1 ,.44,`p/2`]],\n color=black,font=[SYMBOL,9]): \nplots[display]([p1,p2,t1,t2,t3],labels=[``,``],\nxtickmarks=[-1.9099 =`-6`,-1.2732=`-4`,-.63662=`-2`,0=`0`,\n .63662= `2`,1.2732=`4`,1.9099=`6`],\nytickmarks=0,\n font=[SYMBOL,9],view=[- 2.1..2.1,-.6..0.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "p1 := plot([arctan(x),1/(1+x^2)],x=-6.5. .6.5,color=[red,blue]):\np2 := plot([-Pi/2,Pi/2],-6.5..6.5,color=black ,linestyle=3):\nt1 := plots[textplot]([4.5,1.02,`y = arctan x`],color= black):\nt2 := plots[textplot]([[6.5,-.2,`x`],[-.2,1.9,`y`]],color=bla ck):\nt3 := plots[textplot]([[-.35,-1.4,`-p/2`],[-.3,1.4,`p/2`]],\n \+ color=black,font=[SYMBOL,9]):\nplots[display]([ p1,p2,t1,t2,t3],labels=[``,``],xtickmarks=[-6=`-6`,-4=`-4`,-2=`-2`,0=` 0`,2=`2`,4=`4`,6=`6`],\n ytickmarks=[-1=`-1`,1=`1`],view=[-6.5..6.5,- 1.7..1.9]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "read \"D:\\\\Maple7/procdrs/colours.m\";\nread \"D :\\\\Maple7/procdrs/calculus.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "areaplot(1/(1+x^2),x=0..6,color=[purple,cyan,red],are afunction=true,\n tickmarks=[6,2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 25 "Code for triangle picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 400 "p1 := plot( [[0,0],[2,0],[2,1],[0,0]],color=COLOR(RGB,.45,0,.85)):\np2 := plot([0. 61*cos(t),0.61*sin(t),t=0..arctan(.5)],color=black):\np3 := plot([[1.9 ,0],[1.9,.1],[2,.1]],color=black):\nt1 := plots[textplot]([[2.1,0.6,`x `],[0.95,0.6,`a`]],\n font=[HELVETICA,9],color=black):\nt 2 := plots[textplot]([[0.43,0.12,`q`]],font=[SYMBOL,10],color=black): \+ \nplots[display]([p1,p2,p3,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "Circle pictures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 698 "p1 := plot( [cos(t),sin(t),t=-Pi..Pi],color=red):\np2 := plot([[0,0],[0.6,0.8],[0. 6,0]],\n 0..0.6,color=black,linestyle=2):\np3 := plot([[0 .2*cos(t),0.2*sin(t),t=arctan(4/3)..Pi/2],\n [0.6+0.2*cos(t),0.8+0. 2*sin(t),t=arctan(4/3)+Pi..3*Pi/2]],\n color=black):\np4 := plot([ [.55,0],[.55,.05],[.6,.05]],color=black):\nt1 := plots[textplot]([[0.3 ,0.5,`a`],[0.3,-0.05,`x`],[-0.05,1.2,`y`],\n [-0.05,0.95,`A`],[0 .63,0.87,`B`],[0.63,-0.05,`C`],\n [-0.05,-0.05,`O`]],font=[HEL VETICA,9]):\nt2 := plots[textplot]([[0.05,0.15,`q`],[0.55,0.67,`q`]],f ont=[SYMBOL,10]):\nplots[display]([p1,p2,p3,p4,t1,t2],scaling=constrai ned,\n tickmarks=[0,0],view=[-0.1..1.2,-0.1..1.2],labels=[``,``]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 636 "p1 := plot([2*cos(t),2*sin(t),t=-Pi..Pi],color=red): \nr3 := evalf(sqrt(3)): h := evalf(Pi/36):\np2 := plot([[0,0],[1,r3],[ 1,0]],color=black,linestyle=2):\np3 := plot([[seq([0.6*cos(i*h),0.6*si n(i*h)],i=12..18)],\n [seq([1+0.6*cos(i*h),r3+0.6*sin(i*h)],i=48..5 4)]],color=black):\np4 := plot([[.9,0],[.9,.1],[1,.1]],color=black):\n t1 := plots[textplot]([[0.45,1,`2`],[0.5,-0.1,`1`],\n [2.2,-0 .1,`x`],[-0.1,2.2,`y`]],font=[HELVETICA,9],color=black):\nt2 := plots[ textplot]([[0.12,0.45,`p/6`],[0.89,1.35,`p/6`]],font=[SYMBOL,9]):\nplo ts[display]([p1,p2,p3,p4,t1,t2],scaling=constrained,\n tickmark s=[0,0],view=[-.1..2.2,-.1..2.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }