{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Maple Input" -1 256 "Courier" 0 0 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 257 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 258 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 259 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Maroon Emphasis" -1 260 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 260 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 260 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 281 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Norm al" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Indefinite integrals" }}{PARA 0 " " 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 " " 0 "" {TEXT -1 17 "Version: 5.7.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The idea of integration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Preliminary example " }}{PARA 0 "" 0 "" {TEXT 262 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 18 "Find the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "`f '`(x) = 2*x-1; " "6#/-%$f~'G6#%\"xG,&*&\"\"#\"\"\"F'F+F+F+!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(2) = 3;" "6#/-%\"fG6#\"\"#\"\"$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "By inspection any fu nction of the form" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = x^2-x+c;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"F'!\"\"%\"cGF+" }{TEXT -1 15 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 266 1 "c" }{TEXT -1 22 " is a constant, gives " }{XPPEDIT 18 0 " `f '`(x) = 2*x-1;" "6#/-%$f~'G6#%\"xG,&*&\"\"#\"\"\"F'F+F+F+!\"\"" } {TEXT -1 14 ", as required." }}{PARA 0 "" 0 "" {TEXT -1 24 "Since we a re given that " }{XPPEDIT 18 0 "f(2) = 3" "6#/-%\"fG6#\"\"#\"\"$" } {TEXT -1 12 ", (i) gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "3 = 2^2-2+c" "6#/\"\"$,(*$\"\"#F'\"\"\"F'!\"\"%\"cGF(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "which means that " } {XPPEDIT 18 0 "c = 1" "6#/%\"cG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "We conclude that " }{XPPEDIT 18 0 "f(x) = x^2-x+1" "6# /-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"F'!\"\"F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 4 "Note" }{TEXT -1 148 ": The equation (i) describes an infinite family of curves which a ll have the same derivative or gradient function obtained by translati ng the graph " }{XPPEDIT 18 0 "y = x^2-x" "6#/%\"yG,&*$%\"xG\"\"#\"\" \"F'!\"\"" }{TEXT -1 26 " vertically a distance of " }{XPPEDIT 18 0 "a bs(c)" "6#-%$absG6#%\"cG" }{TEXT -1 18 " units upwards if " }{TEXT 267 1 "c" }{TEXT -1 30 " is positive, or downwards if " }{TEXT 268 1 " c" }{TEXT -1 13 " is negative." }}{PARA 0 "" 0 "" {TEXT -1 65 "In the \+ following picture the curve which satisfies the condition " }{XPPEDIT 18 0 "f(2) =3" "6#/-%\"fG6#\"\"#\"\"$" }{TEXT -1 5 " has " }{TEXT 269 1 "y" }{TEXT -1 37 " intercept 1 and is drawn in magenta." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "plot ([seq(x^2-x+c,c=[-2,-1,-0.5,0,0.5,1,2,3])],x=-1..2,\n color=[red,gr een,blue,cyan,coral,magenta,tan,navy]);" }}{PARA 13 "" 1 "" {GLPLOT2D 257 354 354 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$!\"\"\"\"!$F*F*7$$!3[***** \\P&3Y$*!#=$!3w86KTM)*=>F/7$$!3C++Dcx6x()F/$!39(4UIj-\">NF/7$$!3b++]iT DP\")F/$!3>Uw)pIb7C&F/7$$!3A****\\P\"\\J\\(F/$!3?/$=GYA@*oF/7$$!3g*** \\7V0@&oF/$!3'eU`YofFX)F/7$$!3w++DcexdiF/$!3rcM/x#[i#)*F/7$$!3j***\\i+ #QUcF/$!3Qi/L_qR<6!#<7$$!3$****\\i!3%f+&F/$!3!4@=e[6)[7FP7$$!3;++D\"oS :P%F/$!3'Gw'f_Aur8FP7$$!3h*****\\<#)*=PF/$!3sA.K)\\$z*[\"FP7$$!3#***** \\(G3U9$F/$!33j\")p8(=ne\"FP7$$!3Y*****\\-\\r\\#F/$!3k:%p\\cFzo\"FP7$$ !3?+++vGVZ=F/$!3M#pB-jE6y\"FP7$$!3_*****\\(4J@7F/$!3]XM_(*G&H'=FP7$$!3 ;,+]iIKFl!#>$!3Yv(QZFP7$$\"3(R,++]siL#!#?$!3i03L93L-?FP7$$\"3K,++ +!R5'fFap$!33[SS\"*p0c?FP7$$\"3!)***\\P/QBE\"F/$!3>9!35$))H5@FP7$$\"39 ******\\\"o?&=F/$!3!fbx1D04:#FP7$$\"3k++vVb4*\\#F/$!3!*)32!pZX(=#FP7$$ \"3w++DJ'=_6$F/$!3*)G,#>*fZ9AFP7$$\"3#4++vVy!ePF/$!3avfK3jdMAFP7$$\"3' 4+](=WU[VF/$!3/(=>E\\adC#FP7$$\"3s****\\7B>&)\\F/$!334RK2y**\\AFP7$$\" 3w***\\P>:mk&F/$!3g;@\"z))=eC#FP7$$\"3d***\\iv&QAiF/$!3Cr\"H1td]B#FP7$ $\"3j++]PPBWoF/$!3$zm9#>!))f@#FP7$$\"3%*)*****\\Nm'[(F/$!3195*Q/l\")=# FP7$$\"36****\\(yb^6)F/$!3__g>W!eH:#FP7$$\"3')***\\PMaKs)F/$!3fnu\"4xt 86#FP7$$\"3a****\\7TW)R*F/$!37T@*Q*o`c?FP7$$\"3z*****\\@80+\"FP$!3)QO5 ;_'[**>FP7$$\"31++]7,Hl5FP$!3OP(4()3Z/$>FP7$$\"3()**\\P4w)R7\"FP$!3osk Mj%R1'=FP7$$\"3;++]x%f\")=\"FP$!3wEp_LlVw+++!*>=+:FP $!3SR'zo)fj\\7FP7$$\"3-++DE&4Qc\"FP$!3f](f=H4$=6FP7$$\"3=+]P%>5pi\"FP$ !3z#*p8QTt+)*F/7$$\"39+++bJ*[o\"FP$!3Kd9M7@Gg%)F/7$$\"33++Dr\"[8v\"FP$ !3A0G4VvFToF/7$$\"3++++Ijy5=FP$!3b58&3(>R=`F/7$$\"31+]P/)fT(=FP$!3K;1S ?$[oh$F/7$$\"31+]i0j\"[$>FP$!3Iki7\"p@I\">F/7$$\"\"#F*F+-%'COLOURG6&%$ RGBG$\"*++++\"!\")F+F+-F$6$7S7$F($\"\"\"F*7$F-$\"3C'))y'el,\"3)F/7$F3$ \"3'G!z&pO(*3['F/7$F8$\"3!yN7IpW(eZF/7$F=$\"3z&p\"=Pv(y5$F/7$FB$\"39ul M:.CZ:F/7$FG$\"3(GVl&Hs^Pj\")p8(=neF/7$F`o$!3 ]cTp\\cFzoF/7$Feo$!3JBpB-jE6yF/7$Fjo$!31bWBv*G&H')F/7$F_p$!3vbxQZ9!35$))H56FP7$Feq$! 3obvn]_!4:\"FP7$Fjq$!3!*)32!pZX(=\"FP7$F_r$!3*)G,#>*fZ97FP7$Fdr$!3avfK 3jdM7FP7$Fir$!3E(=>E\\adC\"FP7$F^s$!334RK2y**\\7FP7$Fcs$!3g;@\"z))=eC \"FP7$Fhs$!3Cr\"H1td]B\"FP7$F]t$!3$zm9#>!))f@\"FP7$Fbt$!3195*Q/l\")=\" FP7$Fgt$!3__g>W!eH:\"FP7$F\\u$!3fnu\"4xt86\"FP7$Fau$!3!49#*Q*o`c5FP7$F fu$!3')QO5;_'[***F/7$F[v$!3ett4()3Z/$*F/7$F`v$!3xEZYLYR1')F/7$Fev$!3kn #p_LlVw(F/7$Fjv$!3oNBMCc(o*oF/7$F_w$!3C/[37T:7fF/7$Fdw$!3oPK8J`Qs[F/7$ Fiw$!3**['*pktC-PF/7$F^x$!3#RR'zo)fj\\#F/7$Fcx$!3'e](f=H4$=\"F/7$Fhx$ \"35s+j='eE*>Fap7$F]y$\"3oU&ew)yrR:F/7$Fby$\"3y%>2pXA(eJF/7$Fgy$\"3Y*o [\"H!3;o%F/7$F\\z$\"3o$Q*fz;:$Q'F/7$Faz$\"3qNP()3$yp3)F/7$FfzFc[l-Fiz6 &F[[lF+F\\[lF+-F$6$7S7$F($\"3++++++++:FP7$F-$\"3i))y'el,\"38FP7$F3$\"3 G!z&pO(*3[6FP7$F8$\"3!yN7IpW(e(*F/7$F=$\"3z&p\"=Pv(y5)F/7$FB$\"39ulM:. 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*)=FP7$Feq$\"3KWCK\\Z4\\=FP7$Fjq$\"346H*4BXD\"=FP7$F_r$\"35r)z!3S_&y\" FP7$Fdr$\"3YCSn\"pBaw\"FP7$Fir$\"3w73Q2bCa-+v\"FP7$F cs$\"3U$)y376=a,%y\"FP7$Fbt$ \"3$f)*3h&\\$=\"=FP7$Fgt$\"3[ZR!e&>/Z=FP7$F\\u$\"3UKD3Hii))=FP7$Fau$\" 35fy51JYV>FP7$Ffu$\"36O'*QyM^+?FP7$F[v$\"3ki-H6Hbp?FP7$F`v$\"3KFNlO0OR @FP7$Fev$\"3,tIZmMcBAFP7$Fjv$\"3VmdcPCJ5BFP7$F_w$\"3N>:z)e%y3CFP7$Fdw$ \"3Bwm)oYhF^#FP7$Fiw$\"35N+`j_xHEFP7$F^x$\"3ig.78SO]FFP7$Fcx$\"3S\\-93 2p\")GFP7$Fhx$\"3s+j='eE*>IFP7$F]y$\"3Eaew)yrR:$FP7$Fby$\"3[>2pXA(eJ$F P7$Fgy$\"3'*o[\"H!3;oMFP7$F\\z$\"3PQ*fz;:$QOFP7$Faz$\"3dtt)3$yp3QFP7$F fzFbjn-Fiz6&F[[l$\")`B)e)F^[l$\")fqkdF^[l$\")p:#R%F^[l-F$6$7S7$F($\"\" &F*7$F-$\"3i))y'el,\"3[FP7$F3$\"3G!z&pO(*3[YFP7$F8$\"3MN7IpW(eZ%FP7$F= $\"3-q\"=Pv(y5VFP7$FB$\"3UdY`JSsaTFP7$FG$\"3bacHs^Ppc=pw*HFP7$F[q$\"3#>&ff3 I%R%HFP7$F`q$\"3!e)>**o6q*)GFP7$Feq$\"35WCK\\Z4\\GFP7$Fjq$\"346H*4BXD \"GFP7$F_r$\"35r)z!3S_&y#FP7$Fdr$\"3YCSn\"pBaw#FP7$Fir$\"3)H\"3Q2bCaFF P7$F^s$\"3#44wE>-+v#FP7$Fcs$\"3U$)y376=aFFP7$Fhs$\"3wG3PpA%\\w#FP7$F]t $\"33K`y!)>,%y#FP7$Fbt$\"3$f)*3h&\\$=\"GFP7$Fgt$\"3[ZR!e&>/ZGFP7$F\\u$ \"3UKD3Hii))GFP7$Fau$\"3))ey51JYVHFP7$Ffu$\"36O'*QyM^+IFP7$F[v$\"3ki-H 6HbpIFP7$F`v$\"3KFNlO0ORJFP7$Fev$\"3,tIZmMcBKFP7$Fjv$\"3VmdcPCJ5LFP7$F _w$\"3N>:z)e%y3MFP7$Fdw$\"3Bwm)oYhF^$FP7$Fiw$\"35N+`j_xHOFP7$F^x$\"3ig .78SO]PFP7$Fcx$\"3S\\-932p\")QFP7$Fhx$\"3;,j='eE*>SFP7$F]y$\"3saew)yrR :%FP7$Fby$\"3#*>2pXA(eJ%FP7$Fgy$\"3'*o[\"H!3;oWFP7$F\\z$\"3#z$*fz;:$QY FP7$Faz$\"3dtt)3$yp3[FP7$FfzF^do-Fiz6&F[[l$\")!\\DP\"F^[lF`]p$\")viobF ^[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 13 ", a function " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" } {TEXT -1 24 " with the property that " }{XPPEDIT 18 0 "`F '`(x) = f(x) ;" "6#/-%$F~'G6#%\"xG-%\"fG6#F'" }{TEXT -1 14 " is called an " }{TEXT 257 15 "anti-derivative" }{TEXT -1 4 " or " }{TEXT 257 19 "indefinite \+ integral" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " with respect to " }{TEXT 274 1 "x" }{TEXT -1 15 ", and w e write " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x), x) = F(x)+c;" "6#/-%$IntG6$-%\"fG6#%\"xGF*,&-%\"FG6#F*\"\"\"%\"cGF/" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 89 "An indefinite integral i s really an infinite family of functions obtained by varying the " } {TEXT 257 23 "constant of integration" }{TEXT -1 1 " " }{TEXT 265 1 "c " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Examples" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 6 " Since " }{XPPEDIT 18 0 "Diff([x^2],x) = 2*x;" "6#/-%%DiffG6$7#*$%\"xG \"\"#F)*&F*\"\"\"F)F," }{TEXT -1 19 ", it follows that " }{XPPEDIT 18 0 "Int(2*x,x) = x^2+c;" "6#/-%$IntG6$*&\"\"#\"\"\"%\"xGF)F*,&*$F*F( F)%\"cGF)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "Diff([x^3],x) = 3*x^2;" "6#/-%%DiffG6$7#*$%\"xG\"\"$F)* &F*\"\"\"*$F)\"\"#F," }{TEXT -1 19 ", it follows that " }{XPPEDIT 18 0 "Int(3*x^2,x) = x^3+c;" "6#/-%$IntG6$*&\"\"$\"\"\"*$%\"xG\"\"#F)F+,& *$F+F(F)%\"cGF)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then \+ " }{XPPEDIT 18 0 "Int(x^2,x) = x^3/3 + c" "6#/-%$IntG6$*$%\"xG\"\"#F( ,&*&F(\"\"$F,!\"\"\"\"\"%\"cGF." }{TEXT -1 11 ", because " }{XPPEDIT 18 0 "Diff([x^3/3],x) = 1/3;" "6#/-%%DiffG6$7#*&%\"xG\"\"$F*!\"\"F)*& \"\"\"F-F*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^3],x) = 1/3;" "6 #/-%%DiffG6$7#*$%\"xG\"\"$F)*&\"\"\"F,F*!\"\"" }{XPPEDIT 18 0 "``(3*x^ 2)=x^2" "6#/-%!G6#*&\"\"$\"\"\"*$%\"xG\"\"#F)*$F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "Two of the previous examples can be use d to check that " }{XPPEDIT 18 0 "Int(``(x^2+2*x),x) = x^3/3+x^2+c;" "6#/-%$IntG6$-%!G6#,&*$%\"xG\"\"#\"\"\"*&F-F.F,F.F.F,,(*&F,\"\"$F2!\" \"F.*$F,F-F.%\"cGF." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Si nce " }{XPPEDIT 18 0 "Diff([1/x],x) = Diff([x^(-1)],x);" "6#/-%%DiffG6 $7#*&\"\"\"F)%\"xG!\"\"F*-F%6$7#)F*,$F)F+F*" }{XPPEDIT 18 0 "``=-x^(-2 )" "6#/%!G,$)%\"xG,$\"\"#!\"\"F*" }{XPPEDIT 18 0 "``=-1/x^2" "6#/%!G,$ *&\"\"\"F'*$%\"xG\"\"#!\"\"F+" }{TEXT -1 19 ", it follows that " } {XPPEDIT 18 0 "Int(-1/(x^2),x) = 1/x+c;" "6#/-%$IntG6$,$*&\"\"\"F)*$% \"xG\"\"#!\"\"F-F+,&*&F)F)F+F-F)%\"cGF)" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "Int(1/(x^2),x) = -1/x+c" "6#/-%$IntG6$*&\"\"\"F(*$%\"xG \"\"#!\"\"F*,&*&F(F(F*F,F,%\"cGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Rules for integration and some st andard integrals" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 112 "Since the process of differentiation is \+ interchangeable with addition and multiplication by a constant, namely : " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " Diff( [ f(x) + g(x) ],x) = `` " "6#/-%%DiffG6$7#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F,F-F,%!G" }{XPPEDIT 18 0 "Diff([f(x)],x)" "6#-%%DiffG6$7#-%\"fG6#%\"xGF*" }{XPPEDIT 18 0 " ``+ ``" "6#,&%!G\"\"\"F$F%" }{XPPEDIT 18 0 "Diff([g(x)],x)" "6#-%%Dif fG6$7#-%\"gG6#%\"xGF*" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([a*f(x)],x) = a;" "6#/-%%DiffG6$7#*&%\"aG\" \"\"-%\"fG6#%\"xGF*F.F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([f(x)],x )" "6#-%%DiffG6$7#-%\"fG6#%\"xGF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 43 "the same is true for integration, namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(f(x)+g(x)),x) = Int(f(x) ,x)+Int(g(x),x);" "6#/-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F.F/ F.,&-F%6$-F,6#F.F.F/-F%6$-F16#F.F.F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(a*f(x),x)=a*Int(f(x),x)" "6#/- %$IntG6$*&%\"aG\"\"\"-%\"fG6#%\"xGF)F-*&F(F)-F%6$-F+6#F-F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "Differentiation and integratio n are both " }{TEXT 257 16 "linear processes" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 19 "For example, since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^2+x^3],x) = 2*x+3*x^2;" "6#/-%% DiffG6$7#,&*$%\"xG\"\"#\"\"\"*$F*\"\"$F,F*,&*&F+F,F*F,F,*&F.F,*$F*F+F, F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(2*x+3*x^2),x ) = x^2+x^3+c;" "6#/-%$IntG6$-%!G6#,&*&\"\"#\"\"\"%\"xGF-F-*&\"\"$F-*$ F.F,F-F-F.,(*$F.F,F-*$F.F0F-%\"cGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 17 "Similarly, since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff([5*sqrt(x)],x) = 5/(2*sqrt(x));" "6#/-%%DiffG6$7#* &\"\"&\"\"\"-%%sqrtG6#%\"xGF*F.*&F)F**&\"\"#F*-F,6#F.F*!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(5/(2*sqrt(x)),x) = 5*sq rt(x)+c;" "6#/-%$IntG6$*&\"\"&\"\"\"*&\"\"#F)-%%sqrtG6#%\"xGF)!\"\"F/, &*&F(F)-F-6#F/F)F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "The power rule for differentiation, namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x^r],x)=r*x^(r-1)" "6#/-%%Diff G6$7#)%\"xG%\"rGF)*&F*\"\"\")F),&F*F,F,!\"\"F," }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 23 "leads to the following " }{TEXT 257 26 "p ower rule for integration" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^r,x) = x^(r+1)/(r+1)+c;" "6#/-%$IntG6$) %\"xG%\"rGF(,&*&)F(,&F)\"\"\"F.F.F.,&F)F.F.F.!\"\"F.%\"cGF." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r<>-1" "6#0%\"rG,$\"\"\"!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "This rule is easily checked by differentiation. " }}{PARA 0 "" 0 "" {TEXT -1 47 "Note that the rule cannot be used for the case " }{XPPEDIT 18 0 "r = -1;" "6#/%\"rG,$\"\"\"!\"\"" }{TEXT -1 40 ", which would lead t o division by zero. " }}{PARA 0 "" 0 "" {TEXT -1 37 "This does not cau se problems because " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([ln*x],x) = 1/x;" "6#/-%%DiffG6$7#*&%#lnG\"\"\"%\"xGF*F+*&F*F* F+!\"\"" }{XPPEDIT 18 0 "``=x^(-1)" "6#/%!G)%\"xG,$\"\"\"!\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 " ln*x = log[exp(1)]*x;" "6#/*&%#lnG\"\"\"%\"xGF&*&&%$logG6#-%$expG6#F&F &F'F&" }{TEXT -1 61 " is the natural logarithm function, or logarithm \+ to the base " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(1)=Limit( (1+t)^(1/t),t=0)" "6#/-%$expG6#\"\"\"-%&LimitG6$),&F'F'%\"tGF'*&F'F'F- !\"\"/F-\"\"!" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 14 " 2.71828 1828. " }}{PARA 0 "" 0 "" {TEXT -1 13 "Hence, (when " }{TEXT 271 1 "x " }{TEXT -1 15 " is positive): " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "In t(x^(-1),x)=Int(1/x,x)" "6#/-%$IntG6$)%\"xG,$\"\"\"!\"\"F(-F%6$*&F*F*F (F+F(" }{XPPEDIT 18 0 "`` = ln*x+c;" "6#/%!G,&*&%#lnG\"\"\"%\"xGF(F(% \"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([exp(x)],x) = ex p(x);" "6#/-%%DiffG6$7#-%$expG6#%\"xGF+-F)6#F+" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 17 "it follows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x),x)=exp(x)+c" "6#/-%$IntG6$-% $expG6#%\"xGF*,&-F(6#F*\"\"\"%\"cGF." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "More generally, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(exp(a*x),x)=exp(a*x)/a +c" "6#/-%$IntG6$-%$expG6#*& %\"aG\"\"\"%\"xGF,F-,&*&-F(6#*&F+F,F-F,F,F+!\"\"F,%\"cGF," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 272 1 "a" }{TEXT -1 70 " is a non-zero constant, as can easily be checked by differenti ation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The following standard derivatives of trigonometric functions: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Diff([sin *x],x) = cos*x, ``],[Diff([cos*x],x) = -sin*x, ``]);" "6#-%*PIECEWISEG 6$7$/-%%DiffG6$7#*&%$sinG\"\"\"%\"xGF.F/*&%$cosGF.F/F.%!G7$/-F)6$7#*&F 1F.F/F.F/,$*&F-F.F/F.!\"\"F2" }{TEXT -1 11 " " }{XPPEDIT 18 0 "PIECEWISE([Diff([tan*x],x) = sec^2*x, ``],[Diff([sec*x],x) = sec*x* tan*x, ``])" "6#-%*PIECEWISEG6$7$/-%%DiffG6$7#*&%$tanG\"\"\"%\"xGF.F/* &%$secG\"\"#F/F.%!G7$/-F)6$7#*&F1F.F/F.F/**F1F.F/F.F-F.F/F.F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "give rise to the standard inte grals: " }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "PIECEW ISE([Int(cos*x,x) = sin*x+c, ``],[Int(sin*x,x) = -cos*x+c, ``]);" "6#- %*PIECEWISEG6$7$/-%$IntG6$*&%$cosG\"\"\"%\"xGF-F.,&*&%$sinGF-F.F-F-%\" cGF-%!G7$/-F)6$*&F1F-F.F-F.,&*&F,F-F.F-!\"\"F2F-F3" }{TEXT -1 6 " \+ " }{XPPEDIT 18 0 "PIECEWISE([Int(sec^2*x,x) = tan*x+c, ``],[Int(sec*x *tan*x,x) = sec*x+c, ``])" "6#-%*PIECEWISEG6$7$/-%$IntG6$*&%$secG\"\"# %\"xG\"\"\"F.,&*&%$tanGF/F.F/F/%\"cGF/%!G7$/-F)6$**F,F/F.F/F2F/F.F/F., &*&F,F/F.F/F/F3F/F4" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 16 "M ore generally, " }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "PIECEWISE([Int(cos*a*x,x) = sin*a*x/a+c, ``],[Int(sin*a*x,x) = -cos *a*x/a+c, ``]);" "6#-%*PIECEWISEG6$7$/-%$IntG6$*(%$cosG\"\"\"%\"aGF-% \"xGF-F/,&**%$sinGF-F.F-F/F-F.!\"\"F-%\"cGF-%!G7$/-F)6$*(F2F-F.F-F/F-F /,&**F,F-F.F-F/F-F.F3F3F4F-F5" }{TEXT -1 6 " " }{XPPEDIT 18 0 "PI ECEWISE([Int(sec^2*a*x,x) = tan*a*x/a+c, ``],[Int(sec*a*x*tan*a*x,x) = sec*a*x/a+c, ``]);" "6#-%*PIECEWISEG6$7$/-%$IntG6$*(%$secG\"\"#%\"aG \"\"\"%\"xGF/F0,&**%$tanGF/F.F/F0F/F.!\"\"F/%\"cGF/%!G7$/-F)6$*.F,F/F. F/F0F/F3F/F.F/F0F/F0,&**F,F/F.F/F0F/F.F4F/F5F/F6" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 273 1 "a" }{TEXT -1 24 " is a nonzero constant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Summary of standard integrals " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "matrix([[f(x), Int(f(x),x)], [__________, __________________], [` \+ `*x^r*` `, x^(r+1)/(r+1)+c*`, `*r <> -1], [1/x, ln*x+c*`, `*0 < x ], [exp(a*x), exp(a*x)/a+c]]);" "6#-%'matrixG6#7'7$-%\"fG6#%\"xG-%$Int G6$-F)6#F+F+7$%+__________G%3__________________G7$*(%&~~~~~G\"\"\")F+% \"rGF7%$~~~GF70,&*&)F+,&F9F7F7F7F7,&F9F7F7F7!\"\"F7*(%\"cGF7%#,~GF7F9F 7F7,$F7FA7$*&F7F7F+FA2,&*&%#lnGF7F+F7F7*(FCF7FDF7\"\"!F7F7F+7$-%$expG6 #*&%\"aGF7F+F7,&*&-FP6#*&FSF7F+F7F7FSFAF7FCF7" }{TEXT -1 19 " \+ " }{XPPEDIT 18 0 "matrix([[f(x), Int(f(x),x)], [___________, ________________], [sin*a*x, -cos*a*x/a+c], [cos*a*x, sin*a*x/a+c], [ sec^2*a*x, tan*a*x/a+c], [sec*a*x*tan*a*x, sec*a*x/a+c]]);" "6#-%'matr ixG6#7(7$-%\"fG6#%\"xG-%$IntG6$-F)6#F+F+7$%,___________G%1____________ ____G7$*(%$sinG\"\"\"%\"aGF7F+F7,&**%$cosGF7F8F7F+F7F8!\"\"F<%\"cGF77$ *(F;F7F8F7F+F7,&**F6F7F8F7F+F7F8F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "Setting up a Maple procedure to perform basic integration " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 108 "Because differentiation is linear, integration is also l inear. This means that we can use the Maple command " }{TEXT 256 6 "de fine" }{TEXT -1 36 " to obtain an integration procedure " }{TEXT 0 4 " intg" }{TEXT -1 130 " which will work for any polynomials and indeed f or any expression which is a sum of terms of the form \"constant times a power of " }{TEXT 283 1 "x" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We incorporate the follow ing basic properties." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 278 9 "linearity" }{TEXT -1 15 ": " }{XPPEDIT 18 0 "intg(fx+gx,x) = intg(fx,x)+intg(gx,x);" "6#/-%%intgG6$,&%#fxG\" \"\"%#gxGF)%\"xG,&-F%6$F(F+F)-F%6$F*F+F)" }{TEXT -1 46 " \+ \n " }{XPPEDIT 18 0 "intg(c*fx,x) = c*in tg(fx,x);" "6#/-%%intgG6$*&%\"cG\"\"\"%#fxGF)%\"xG*&F(F)-F%6$F*F+F)" } }{PARA 15 "" 0 "" {TEXT 279 14 "the power rule" }{TEXT -1 6 ": " } {XPPEDIT 18 0 "intg(x^r) = x^(r+1)/(r+1);" "6#/-%%intgG6#)%\"xG%\"rG*& )F(,&F)\"\"\"F-F-F-,&F)F-F-F-!\"\"" }{TEXT -1 6 " , if " }{XPPEDIT 18 0 "r <> -1;" "6#0%\"rG,$\"\"\"!\"\"" }}{PARA 15 "" 0 "" {TEXT 280 12 " special case" }{TEXT -1 11 ": " }{XPPEDIT 18 0 "intg(1/x,x) = ln(x);" "6#/-%%intgG6$*&\"\"\"F(%\"xG!\"\"F)-%#lnG6#F)" }}{PARA 15 " " 0 "" {TEXT 281 9 "constants" }{TEXT -1 15 ": " } {XPPEDIT 18 0 "intg(1,x) = x;" "6#/-%%intgG6$\"\"\"%\"xGF(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Note that we do n ot include a constant of integration." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "define(intg,linear,\nco nditional(intg((x::name)^(r::realcons),x::name)=\n x^(r+1)/(r+1),r<> -1),\nintg((x::name)^(-1),x::name)=ln(x),\nintg(1,x::name)=x);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 8 "Examples " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fx := 2*x^3+x^2-7*x-2+1/x:\nInt(fx,x)=intg(fx ,x)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,,*&\"\"#\"\"\")% \"xG\"\"$F*F**$)F,F)F*F**&\"\"(F*F,F*!\"\"F)F2*&F*F*F,F2F*F,,.*&F)F2F, \"\"%F**&F-F2F,F-F**(F1F*F)F2F,F)F2*&F)F*F,F*F2-%#lnG6#F,F*%\"cGF*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fx := 1/sqrt(2*x)-1/(2*x):\nInt(fx,x)=intg(fx,x)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,&*(\"\"#!\"\"F)#\"\"\"F)%\"xG#F*F)F, *&F,F,*&F)F,F-F,F*F*F-,(*&F)F+F-F+F,*&#F,F)F,-%#lnG6#F-F,F*%\"cGF," }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The expr ession " }{XPPEDIT 18 0 "x*(2*x-3)^7" "6#*&%\"xG\"\"\"*$,&*&\"\"#F%F$F %F%\"\"$!\"\"\"\"(F%" }{TEXT -1 35 " can be integrated with respect to " }{TEXT 284 1 "x" }{TEXT -1 26 " if it is expanded first. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fx := x*(2*x-3)^7:\nInt(fx,x)=intg(fx,x)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"),&*&\"\"#F)F(F)F)\"\"$!\"\"\"\" (F)F(,&-%%intgGF&F)%\"cGF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "fx := x*(2*x-3)^7:\nInt(fx,x)=Int(e xpand(fx),x);\n``=intg(expand(fx),x)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"),&*&\"\"#F)F(F)F)\"\"$!\"\"\"\"(F)F(-F%6 $,2*&\"$G\"F))F(\"\")F)F)*&\"%W8F))F(F0F)F/*&\"%[gF))F(\"\"'F)F)*&\"&? ^\"F))F(\"\"&F)F/*&\"&!oAF))F(\"\"%F)F)*&\"&7/#F))F(F.F)F/*&\"&1-\"F)) F(F-F)F)*&\"%(=#F)F(F)F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,4*& #\"$G\"\"\"*\"\"\"*$)%\"xGF)F*F*F**&\"$o\"F*)F-\"\")F*!\"\"*&\"$k)F*)F -\"\"(F*F**&\"%?DF*)F-\"\"'F*F2*&\"%OXF*)F-\"\"&F*F**&\"%.^F*)F-\"\"%F *F2*&\"%-MF*)F-\"\"$F*F**&#\"%(=#\"\"#F**$)F-FJF*F*F2%\"cGF*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "The Maple integration c ommand: " }{TEXT 0 3 "int" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 33 "The syntax for the Maple command " }{TEXT 256 3 "int" }{TEXT -1 90 " for performing indefinite integratio n is similar to that of the differentiation command " }{TEXT 0 4 "dif f" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 52 "The following comman d finds the indefinite integral " }{XPPEDIT 18 0 "Int(``(2*x^3+x^2-7*x -2+1/x),x);" "6#-%$IntG6$-%!G6#,,*&\"\"#\"\"\"*$%\"xG\"\"$F,F,*$F.F+F, *&\"\"(F,F.F,!\"\"F+F3*&F,F,F.F3F,F." }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 110 "As with the bare-bones integration procedure defined i n the last section, no constant of integration is given." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(2 *x^3+x^2-7*x-2+1/x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\" \"%\"\"\"#F(\"\"#*$)F&\"\"$F(#F(F-*$)F&F*F(#!\"(F*F&!\"#-%#lnG6#F&F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The in ert form " }{TEXT 0 4 "Int " }{TEXT -1 15 "of the command " }{TEXT 256 3 "int" }{TEXT -1 101 " is useful for checking that you have used \+ the correct Maple input for the integral you wish to find." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int( 2*x^3+x^2-7*x-2+1/x,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$,,*$)%\"xG\"\"$\"\"\"\"\"#*$)F)F,F+F+F)!\"(!\"#F+*&F+F+F)!\" \"F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\"%\"\"\"#F(\"\" #*$)F&\"\"$F(#F(F-*$)F&F*F(#!\"(F*F&!\"#-%#lnG6#F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "An initial value problem can be solved by introducing a constant of integration. " }}{PARA 0 "" 0 "" {TEXT -1 38 "For example, let's find the function " }{XPPEDIT 18 0 "y = y(x); " "6#/%\"yG-F$6#%\"xG" }{TEXT -1 32 " which satisfies the conditions: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = x^2+2*x; " "6#/*&%#dyG\"\"\"%#dxG!\"\",&*$%\"xG\"\"#F&*&F,F&F+F&F&" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "y(1) = 5;" "6#/-%\"yG6#\"\"\"\"\"&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fx := int(x^2+2*x,x) + c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,(*&#\"\"\"\"\"$F(*$)%\"xGF)F(F(F(*$)F,\"\"#F(F(% \"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can use the initial condition to find the constant " }{TEXT 277 1 "c" }{TEXT -1 64 ", and give the solution both as an expression \+ and as a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 88 "eq := subs(\{x=1,y=5\},y=fx);\nc1 := solve(e q,c);\nfx := subs(c=c1,fx);\nf := unapply(fx,x);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#eqG/\"\"&,&#\"\"%\"\"$\"\"\"%\"cGF+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#c1G#\"#6\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,(*&#\"\"\"\"\"$F(*$)%\"xGF)F(F(F(*$)F,\"\"#F(F(#\"#6F)F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,(*&#\"\"\"\"\"$F/*$)9$F0F/F/F/*$)F3\"\"#F/F/#\"#6F0F/F(F(F(" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We can \+ easily check the result.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(f(x),x);\nf(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG \"\"#\"\"\"F(F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exa mple 1 " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x^3-5 *sqrt(x))/x,x) = Int(``(x^2-5*sqrt(x)/x),x);" "6#/-%$IntG6$*&,&*$%\"xG \"\"$\"\"\"*&\"\"&F,-%%sqrtG6#F*F,!\"\"F,F*F2F*-F%6$-%!G6#,&*$F*\"\"#F ,*(F.F,-F06#F*F,F*F2F2F*" }{XPPEDIT 18 0 "`` = Int(``(x^2-5*x^(-1/2)), x);" "6#/%!G-%$IntG6$-F$6#,&*$%\"xG\"\"#\"\"\"*&\"\"&F.)F,,$*&F.F.F-! \"\"F4F.F4F," }{XPPEDIT 18 0 "``=x^3/3-10*x^(1/2)+c" "6#/%!G,(*&%\"xG \"\"$F(!\"\"\"\"\"*&\"#5F*)F'*&F*F*\"\"#F)F*F)%\"cGF*" }{XPPEDIT 18 0 "``=x^3/3-10*sqrt(x)+c" "6#/%!G,(*&%\"xG\"\"$F(!\"\"\"\"\"*&\"#5F*-%%s qrtG6#F'F*F)%\"cGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int((x^3-5*sqrt(x))/x,x); \nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*$)%\"xG\" \"$\"\"\"F,*&\"\"&F,F*#F,\"\"#!\"\"F,F*F1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#5\"\"\"%\"xG#F&\"\"#!\"\"*&\"\"$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 10 "Example 2 " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x*sqrt(x)+Pi/x),x) = Int(``(x^(3/2)+Pi/x),x);" "6#/-%$IntG6$-%!G6#,&*&%\"xG\"\"\"-%%sqrtG6 #F,F-F-*&%#PiGF-F,!\"\"F-F,-F%6$-F(6#,&)F,*&\"\"$F-\"\"#F3F-*&F2F-F,F3 F-F," }{XPPEDIT 18 0 "``=2/3" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(5/2)+Pi*ln*x+c;" "6#,()%\"xG*&\"\"&\"\"\" \"\"#!\"\"F(*(%#PiGF(%#lnGF(F%F(F(%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(x *sqrt(x)+Pi/x,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$,&*$)%\"xG#\"\"$\"\"#\"\"\"F-*&%#PiGF-F)!\"\"F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#\"\"\"\"\"&!\"\"%\"xG#F'F%F&*&%#PiGF&-%#lnG 6#F)F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "Int(``(sin*2*x-cos(x/2)+exp(-3*x)),x) = -cos*2*x/2-2 *sin(x/2)-1/3;" "6#/-%$IntG6$-%!G6#,(*(%$sinG\"\"\"\"\"#F-%\"xGF-F--%$ cosG6#*&F/F-F.!\"\"F4-%$expG6#,$*&\"\"$F-F/F-F4F-F/,(**F1F-F.F-F/F-F.F 4F4*&F.F--F,6#*&F/F-F.F4F-F4*&F-F-F:F4F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-3*x)+c" "6#,&-%$expG6#,$*&\"\"$\"\"\"%\"xGF*!\"\"F*%\"cGF* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "Int(sin(2*x)-cos(x/2)+exp(-3*x),x);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(-%$sinG6#,$*&\"\"#\"\" \"%\"xGF-F-F--%$cosG6#,$*&F,!\"\"F.F-F-F4-%$expG6#,$*&\"\"$F-F.F-F4F-F ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&-%$cosG6#,$*&F' F&%\"xGF&F&F&!\"\"*&F'F&-%$sinG6#,$*&F'F.F-F&F&F&F.*&#F&\"\"$F&-%$expG 6#,$*&F7F&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 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(tan*x*(sec*x+tan*x),x)=Int(``(tan*x*sec*x +tan^2*x),x)" "6#/-%$IntG6$*(%$tanG\"\"\"%\"xGF),&*&%$secGF)F*F)F)*&F( F)F*F)F)F)F*-F%6$-%!G6#,&**F(F)F*F)F-F)F*F)F)*&F(\"\"#F*F)F)F*" } {XPPEDIT 18 0 "``=Int(``(sec*x*tan*x+sec^2*x-1),x)" "6#/%!G-%$IntG6$-F $6#,(**%$secG\"\"\"%\"xGF-%$tanGF-F.F-F-*&F,\"\"#F.F-F-F-!\"\"F." } {XPPEDIT 18 0 "``=sec*x+tan*x-x+c" "6#/%!G,**&%$secG\"\"\"%\"xGF(F(*&% $tanGF(F)F(F(F)!\"\"%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(tan(x)*(sec(x) +tan(x)),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*& -%$tanG6#%\"xG\"\"\",&-%$secGF)F+F'F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"\"F%-%$cosG6#%\"xG!\"\"F%-%$tanGF(F%F)F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 " " 0 "" {TEXT 275 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "Given that " }{XPPEDIT 18 0 "`f '`(x) = 3*cos*x+5*sin*x; " "6#/-%$f~'G6#%\"xG,&*(\"\"$\"\"\"%$cosGF+F'F+F+*(\"\"&F+%$sinGF+F'F+ F+" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "f(0)=4" "6#/-%\"fG6#\"\"!\"\" %" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 14 " has the form " }{XPPEDIT 18 0 "Int(``(3*cos*x+5*sin*x),x)" "6#-%$IntG6$-%!G6#,&*( \"\"$\"\"\"%$cosGF,%\"xGF,F,*(\"\"&F,%$sinGF,F.F,F,F." }{TEXT -1 5 ", \+ so " }{XPPEDIT 18 0 "f(x)=3*sin*x-5*cos*x+c" "6#/-%\"fG6#%\"xG,(*(\"\" $\"\"\"%$sinGF+F'F+F+*(\"\"&F+%$cosGF+F'F+!\"\"%\"cGF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The condition " }{XPPEDIT 18 0 "f(0) =4" "6#/-%\"fG6#\"\"!\"\"%" }{TEXT -1 15 ", implies that " }{XPPEDIT 18 0 "4=-5+c" "6#/\"\"%,&\"\"&!\"\"%\"cG\"\"\"" }{TEXT -1 10 ", so tha t " }{XPPEDIT 18 0 "c=9" "6#/%\"cG\"\"*" }{TEXT -1 8 ". Hence " } {XPPEDIT 18 0 "f(x)=3*sin*x-5*cos*x+9" "6#/-%\"fG6#%\"xG,(*(\"\"$\"\" \"%$sinGF+F'F+F+*(\"\"&F+%$cosGF+F'F+!\"\"\"\"*F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "int(3*cos(x)+5*sin(x),x) + c;\nsubs(\{x=0,y=4\},y=%);\nc=solve(%,c );\n'f'(x) = subs(%,fx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"$ \"\"\"-%$sinG6#%\"xGF&F&*&\"\"&F&-%$cosGF)F&!\"\"%\"cGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/\"\"%,(*&\"\"$\"\"\"-%$sinG6#\"\"!F(F(*&\"\"&F( -%$cosGF+F(!\"\"%\"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG\"\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&\"\"$\"\"\"-%$si nGF&F+F+*&\"\"&F+-%$cosGF&F+!\"\"\"\"*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 4 "No te" }{TEXT -1 59 ": This question involves solving the differential eq uation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=3*co s*x+5*sin*x" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*(\"\"$F&%$cosGF&%\"xGF&F&*( \"\"&F&%$sinGF&F-F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " subject to the condition that " }{XPPEDIT 18 0 "y=4" "6#/%\"yG\"\"%" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 70 "This differential equation can be s olved by using the Maple procedure " }{TEXT 0 6 "dsolve" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "f := 'f':\nde := diff(f(x),x)=3*cos(x)+5*sin(x);\nic \+ := f(0)=4;\ndsolve(\{de,ic\},f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/-%%diffG6$-%\"fG6#%\"xGF,,&*&\"\"$\"\"\"-%$cosGF+F0F0*&\"\"&F0 -%$sinGF+F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"fG6#\"\"! \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&\"\"$\"\"\" -%$sinGF&F+F+*&\"\"&F+-%$cosGF&F+!\"\"\"\"*F+" }}}{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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 40 "Find the following indefinite integrals. " }}{PARA 0 "" 0 "" {TEXT -1 8 " (a) " }{XPPEDIT 18 0 "Int(``(3*x^4 -1/x^2+sqrt(x)),x)" "6#-%$IntG6$-%!G6#,(*&\"\"$\"\"\"*$%\"xG\"\"%F,F,* &F,F,*$F.\"\"#!\"\"F3-%%sqrtG6#F.F,F." }{TEXT -1 20 " (b ) " }{XPPEDIT 18 0 "Int((x^3-5)/x^2,x)" "6#-%$IntG6$*&,&*$%\"xG\"\"$ \"\"\"\"\"&!\"\"F+*$F)\"\"#F-F)" }{TEXT -1 17 " (c) " } {XPPEDIT 18 0 "Int((x+1)^3/x,x)" "6#-%$IntG6$*&,&%\"xG\"\"\"F)F)\"\"$F (!\"\"F(" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " (d) " }{XPPEDIT 18 0 "Int(``(2*exp(-3*x)-exp(x/ 4)),x)" "6#-%$IntG6$-%!G6#,&*&\"\"#\"\"\"-%$expG6#,$*&\"\"$F,%\"xGF,! \"\"F,F,-F.6#*&F3F,\"\"%F4F4F3" }{TEXT -1 11 " (e) " }{XPPEDIT 18 0 "Int(``(cos*Pi*x-sin(x/3)),x)" "6#-%$IntG6$-%!G6#,&*(%$cosG\"\"\" %#PiGF,%\"xGF,F,-%$sinG6#*&F.F,\"\"$!\"\"F4F." }{TEXT -1 8 " (f) " }{XPPEDIT 18 0 "Int(``(tan^2*x+2),x)" "6#-%$IntG6$-%!G6#,&*&%$tanG\"\" #%\"xG\"\"\"F.F,F.F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "3/5" "6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^5+1/x+2/3" "6#,(*$%\"xG\"\"&\"\"\"*&F'F'F%! \"\"F'*&\"\"#F'\"\"$F)F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2)+c" " 6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(%\"cGF(" }{TEXT -1 9 " (b) " } {XPPEDIT 18 0 "x^2/2+5/x+c" "6#,(*&%\"xG\"\"#F&!\"\"\"\"\"*&\"\"&F(F%F 'F(%\"cGF(" }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "x^3/3+3/2" "6#,&*&% \"xG\"\"$F&!\"\"\"\"\"*&F&F(\"\"#F'F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+3*x+ln*x+c;" "6#,**$%\"xG\"\"#\"\"\"*&\"\"$F'F%F'F'*&%#lnGF'F%F'F '%\"cGF'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " (d) " } {XPPEDIT 18 0 " -2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-3*x)-4*exp(x/4)+c" "6#,(-%$expG6#,$*&\"\"$\"\"\"% \"xGF*!\"\"F**&\"\"%F*-F%6#*&F+F*F.F,F*F,%\"cGF*" }{TEXT -1 9 " (e) \+ " }{XPPEDIT 18 0 "sin*Pi*x/Pi+3*cos(x/3)+c" "6#,(**%$sinG\"\"\"%#PiG F&%\"xGF&F'!\"\"F&*&\"\"$F&-%$cosG6#*&F(F&F+F)F&F&%\"cGF&" }{TEXT -1 8 " (f) " }{XPPEDIT 18 0 "tan*x+x+c" "6#,(*&%$tanG\"\"\"%\"xGF&F&F' F&%\"cGF&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 30 " ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "____________ __________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 44 "The gradient of a certain curve is given by " }{XPPEDIT 18 0 "dy/d x=1/sqrt(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&-%%sqrtG6#%\"xGF(" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "If the curve passes th rough the point" }{XPPEDIT 18 0 " ``(4,0)" "6#-%!G6$\"\"%\"\"!" } {TEXT -1 34 ", find the equation of the curve. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2*sqrt(x)-4" "6#/%\"yG,&*& \"\"#\"\"\"-%%sqrtG6#%\"xGF(F(\"\"%!\"\"" }{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 30 " ______________________________" }}{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 30 "______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 33 " in each of the following cases. " }}{PARA 0 "" 0 "" {TEXT -1 11 " (a) " }{XPPEDIT 18 0 "`f '`(x) = 12*x^2-24*x+1;" " 6#/-%$f~'G6#%\"xG,(*&\"#7\"\"\"*$F'\"\"#F+F+*&\"#CF+F'F+!\"\"F+F+" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "f(1)=-2" "6#/-%\"fG6#\"\"\",$\"\"#!\" \"" }{TEXT -1 14 ", (b) " }{XPPEDIT 18 0 "`f '`(x) = 3*sqrt(x) -1/sqrt(x);" "6#/-%$f~'G6#%\"xG,&*&\"\"$\"\"\"-%%sqrtG6#F'F+F+*&F+F+-F -6#F'!\"\"F2" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(1)=2" "6#/-%\"fG6#\" \"\"\"\"#" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 " (c) \+ " }{XPPEDIT 18 0 "`f '`(x) = 2*cos*x-sin*x+1;" "6#/-%$f~'G6#%\"xG,(*( \"\"#\"\"\"%$cosGF+F'F+F+*&%$sinGF+F'F+!\"\"F+F+" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "f(0)=2" "6#/-%\"fG6#\"\"!\"\"#" }{TEXT -1 14 ", \+ (d) " }{XPPEDIT 18 0 "`f ''`(x) = 20*x^3-10;" "6#/-%%f~''G6#%\"xG,&*& \"#?\"\"\"*$F'\"\"$F+F+\"#5!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(1 )=1" "6#/-%\"fG6#\"\"\"F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`f '`(1) = -5;" "6#/-%$f~'G6#\"\"\",$\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 7 " (a) " }{XPPEDIT 18 0 "f(x) = 4*x^3-12*x ^2+x+5" "6#/-%\"fG6#%\"xG,**&\"\"%\"\"\"*$F'\"\"$F+F+*&\"#7F+*$F'\"\"# F+!\"\"F'F+\"\"&F+" }{TEXT -1 10 " (b) " }{XPPEDIT 18 0 "f(x)=2*x ^(3/2)-2*sqrt(x)+2" "6#/-%\"fG6#%\"xG,(*&\"\"#\"\"\")F'*&\"\"$F+F*!\" \"F+F+*&F*F+-%%sqrtG6#F'F+F/F*F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " (c) " }{XPPEDIT 18 0 "f (x) = 2*sin*x+cos*x+x+1" "6#/-%\"fG6#%\"xG,**(\"\"#\"\"\"%$sinGF+F'F+F +*&%$cosGF+F'F+F+F'F+F+F+" }{TEXT -1 8 " (d) " }{XPPEDIT 18 0 "f(x) =x^5-5*x^2+5" "6#/-%\"fG6#%\"xG,(*$F'\"\"&\"\"\"*&F*F+*$F'\"\"#F+!\"\" F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 40 "Find the following indefinite integrals: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "(a) \+ " }{XPPEDIT 18 0 "Int(x^5+3*sqrt(x)-1/(x^3),x);" "6#-%$IntG6$,(*$%\"xG \"\"&\"\"\"*&\"\"$F*-%%sqrtG6#F(F*F**&F*F**$F(F,!\"\"F2F(" }}{PARA 0 " " 0 "" {TEXT -1 42 "__________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "____________________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "(b) " }{XPPEDIT 18 0 "Int(sec(5*x)^2-sin(2*x)/3+cos (Pi*x),x);" "6#-%$IntG6$,(*$-%$secG6#*&\"\"&\"\"\"%\"xGF-\"\"#F-*&-%$s inG6#*&F/F-F.F-F-\"\"$!\"\"F6-%$cosG6#*&%#PiGF-F.F-F-F." }}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________________" }}{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 42 "_______________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "(c) " }{XPPEDIT 18 0 "Int(Sum(x^i,i = 0 .. 10),x);" "6#-%$IntG 6$-%$SumG6$)%\"xG%\"iG/F+;\"\"!\"#5F*" }}{PARA 0 "" 0 "" {TEXT 257 4 " Note" }{TEXT -1 63 ": You can see what is to be integrated by using th e commands: " }{TEXT 256 17 "Sum(x^i,i=0..10);" }{TEXT -1 108 "\n \+ \+ " }{TEXT 256 9 "value(%);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________ ________" }}{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 42 "____________ ______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 43 "Solve the following initial value problems." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(x),y(4) = 0;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&-% %sqrtG6#%\"xGF(/-%\"yG6#\"\"%\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "_________________________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "__________ __________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "dy/dx = s in(x)+cos(x),y(0) = 0;" "6$/*&%#dyG\"\"\"%#dxG!\"\",&-%$sinG6#%\"xGF&- %$cosG6#F-F&/-%\"yG6#\"\"!F5" }}{PARA 0 "" 0 "" {TEXT -1 52 "_________ ___________________________________________" }}{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 52 "_______________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "( c) " }{XPPEDIT 18 0 "dy/dx = Sum(sin(n*x),n = 1 .. 8),y(0) = 0;" "6$/ *&%#dyG\"\"\"%#dxG!\"\"-%$SumG6$-%$sinG6#*&%\"nGF&%\"xGF&/F0;F&\"\")/- %\"yG6#\"\"!F9" }}{PARA 0 "" 0 "" {TEXT -1 52 "_______________________ _____________________________" }}{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 52 "____________________________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }