数学分析教案

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Ḅ;⚪t?uv/(X)B:(X)=᝞uvS஺)BV(0};⚪Ḅ:;⚪t?uv~'(X),B/(X)=᝞uvV஺)BS}:;⚪tMᦪᑖO+⌕஺i1.1ᦪᙠIjᨵi$ᙠᦪF(x),Vxelᨵ?F'(x)=/(x)ᑣᦪF(x)tᦪ/(x)ᙠIjḄOᦪ஺ᐵi1.1ᨵ᝞;⚪?1ஹ/(x)Ḅᦪ(᝞ᙠ)ᨵO᝞(x)t/(x)ḄOᦪ$ᑣ/(x)+C¡tḄOᦪᓽ/(x)Ḅᦪᨵ£ᦪO஺2ஹf(x)Ḅ¥OᦪপஹG(x)ᨵ¨ᐵ©஺᝞(x)G(x)ªt/(x)Ḅᦪ$ᑣF(x)=G(x)+C,ᓽ¬®¯°O±ᦪ஺²j$ᨵᳮ1.1ᳮ1.1᝞f(x)t/(X)ḄOᦪ$ᑣf(x)+C³⊤R“X)Ḅᡠᨵᦪ஺i1.2/(x)ᙠIjḄᡠᨵᦪ¶/(x)Ḅᑖ$·[j7(x)"x஺ᐸ»S¶ᑖ¼$/(x)¶½ᦪ$/(x)dx¶½⊤¿

2*$X¶ᑖÀÁ஺=᝞jx2Jx=^-x3Jx+c,1(cosxdx=sinx+cஹᑖḄឋK1ஹ(J/(x)dx)'=/(x),dJ7(x)dx=/(x)dx2ஹ”(x)dx=/(x)+c,W(x)=஻x)+c3ஹJ"(x)
g(x)]dx=J7(x)dx
Jg(x)dx4ஹ^kf(x)dx=uஹ'(ᑖ⊤1ஹ^kdx=kx+c2ஹ[x",dx=-^—xm+'+c3ஹj—=InIxI+c.4ஹl*6fxdx=---Fc,je'dx=€x+cJIn(2J5ஹjsinxdx=-cosx+c6ஹJcosxdx=sinx+c7ஹjsec2xdx=tanx+c8ஹJcsc2xdx=-cotx+c9ஹjtanxsecx^=secx+c10ஹ(cotxcscxrfx=-csc+c11ஹarcsinx+cdx12ஹ---7=arctanx+c1+x2=1Bᑖ

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13£/(x)Jx=Oឋz2(ឋឋz),S°(x),±(x)eR[a,ᑗ,ᑣCD¶6·6ᦪᓰ/*)+»2¼½€ᡂ4f],fᓄ°(x)+kf(x)dx]=ᓃf°(x)dx+k^f(x)dx2222Kᑭᵨ
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14ñTᙠ஻w(0,1),å/(஻)+஻/'(஻)=0.ᑖ᪆(ôõᦪF(x)=#(x),ᵨ
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16ff(x)dx--^f(~)l<—.@nnnk=2Pஹklm_re)Ḅ,-st1ᑭᵨÒÓSᦪÔÓUᑖ¢i8ᳮÕÖ8ᳮ஺Iz1¹/(X)ᙠ¡0,1A¯°ᙠ(0,1)'×஺
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19£/(X)JX=F(X)I>F0)-F(«)-"Ä5ᑖÅᐭdr▲15ᑖrdx11+¶Ä®ᜓ.][71--arctanxL=arctanx\"arctan0=—2°4nn11Jtsin2^dx=£——dx=—(x-sinx)I\?Ë-SÍ)-0]=ᐔ72⚪/1ᑖ34ᦪḄ5ᑖ¦Î(ᑭᵨᑖz{}±ឋ2x,0

20Ù³jmax{x,x1]dx=fxdx+jx2Jx=^x21:I28_J__17ߟ/+1"]"7.⚪/2,â78ᑖḄ᩽▲¦Î(ᑭᵨᑖãä5ᳮ1᩽▲lim2"ᵯX.x(Aு0)”ߟ>ooJnxÄ..r+isinx,„r+*dxlim-----=dxhmsinm—“T8Jnxn->ooJn/=limsinInxI":=limsin[ln(஻+2)-In/?]n—»oo“Too=limln(l+—)sin=lim—sin=0.n—>oo<஻">8M⚪/39᩽▲¦Î(ᦋᑏ9᩽▲nlimê஻ë777nnᑣ¾᩽▲=limV/(-)-=ffMdx.n1᩽▲limYÄlimX2)=limV———ᣚí=x,“-8î+ᓃ—Wj+(?nnn'=dxfI12dx-arctanxI(=71n72᩽▲lim---------........(p>0)“T8YV21஻+2'+.஻஻2SkpÄlim----------(=lim)——r஻f°஻“+஻f°£஻P"

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52lim=limIllxl=Z-lxl"T8"f8a“(1)0ᦪᦣᦈR=.(2)2=0,xjlx1=0<1,!"ᦪ#ᦈ,R=0.$1%&ᑡ(ᦪḄᦈ*(1)(2)£(,1-3/(஻+2)3"3n82nবZ(-ir'^r-n=lJ-L4ஹᗩᦪḄᑖ᪆ឋ:.;ᳮ2=>ᦪḄ?@ᦪAs(%)cᦈAR,R>0,71=0(1)S(X)ᙠ(-R,+R)GH;(2)5(%)ᙠ(-R,R)ᑁLMNOL⌲⚗MᑖᓽVxe(-R,R),UVW4஺/WLZ-পn=0"=0஻ᓝ1(3)s(x)ᙠ(ߟH,R)ᑁᨵGH`ᦪNOL⌲⚗%`ᓽ00O0s'(x)(2)n=0w=0(4)ᦪ(1)b(2)bcᦪᨵdeḄᦈ.$2%&ᑡfᦪḄ?@ᦪ.(1)Sc-ir1—ফ£(஻+1)l"n=l஻n=0mn(1)᧕pᦈA1.=

53s(x)=£(Trᡈxe(-l,l)“=1<ᑣ“x)=ᳮ,1-(u="=1஻n=1=1—x+x2,•-+(ߟ1)-IX'~I+-••=---,ᦑ1+X{ns'(x)dt=£-^ߟ=ln(l+x)ᓽ5(x)-5(0)=ln(l+x)s(x)=ln(l+x)xe(-l,l)(2)ᦈA1,=/(x)=£(n+l)x"xe(-l,l)”=0ᑣ(/«)W=£(("+1)஻W=££~”=0Rn=0...............X.1n1(3)3n1ஹᦈᦈᦈ*Ḅ%ᦈbᦈ*.2ஹ?@ᦪḄᑖ᪆ឋ:?@ᦪḄ%.@ᦪḄᦪᦪ5⁚ᵨ4ᦟ᛻ᦟᦟḄ¡¢@ᦪᡂ>ᦪḄ¤¥¦§¨¦§ᣴḄᑁªஹ⁚ஹ¤

54ᣴ,ஹ@ᦪḄᦪᑁ«/¬®u¯¬᪍®±²/¬ᦪu¯¬᪍ᦪªᖧ/´%µ¶®/¬ᦪᡈu¯¬᪍ᦪḄᐙ⌕ᩩº᯿¼ᦪ®.4ஹ½¾@ᦪḄ>ᦪ⁚1n¿`5ÀÁᵨḄ®⁚2n$⚪çJ½¾Æᣚ1ÈᑖMᑖ⁚3nᚪÊஹ᜜ÌÍP491ஹ(1)ஹফஹ(3)3ஹপஹ(2)ÐὃᦻÓᦪᑖ᪆Ô´&µᑢ᳝×ØP90-94>Ùᑁªᜓ,ஹ@ᦪḄᦪ,ÛᭆÝᦪ஺µ+Þ´x.x஺µ+ß´,ᑗ+…+âã1!2!n\´X-X஺µ"H---´1µåA/´Xµᙠ/ᜐḄ/¬ᦪçᓣ4°å/´Oµ+Sx+ᦋ,…ì…´2µ1!2!஻ãA/´xµḄu¯¬᪍ᦪ.ᦪ´1µᡈ´2µíîᦈ/´xµ,᝞ðᦪ´1µ´ᡈ´2µµᙠñµ´ᡈX0=0µ▬óᦈNOᦈ“Xµ,ᑣå/´xµ¶®(ᦪ´1µ´ᡈ´2µµ.

55;ᳮ1=@ᦪ/(x)ᙠ¦ôḄõÀ⚞*v(%o)ᑁᐹᨵ▤`ᦪᑣùḄ/¬ᦪ(1)ᙠ஻(%0)ᑁᦈoVxev(x),limR“(x)=0.U0M-»00;ᳮ2ú@ᦪ/(x)ᙠû°Ḅõ⚞*ᑁ¶®ᦪ./(x)=Z%(x-Xo)"n=0ᑣᐸ¼ᦪ4=£2(஻=0],2…)n\ýþnÿ஻x)ᙠ0ḄᓽU(x0)ᑁᦪ/(x)=£a„(x-x)n,0஻=0ᑣ⌲⚗ᑖ:/(%)=/'(%)=%Wo)=2%fm(x=n\ajaon—)n\ஹᦪḄᦪ!.#$%⌕!.x2n8nxX1ஹev=l+x+—+•••+—+•••2!n!஻=0'3v5.+1+…=£(,1)஻2ஹsinx=x--4--++(T)஻3!5!(+1)!஻=0(2/1+1)!82nY3ஹcosX=1--——I----1-(-1)M+S(-1)(W2!(2/1)!4ஹln(l+x)=x——+—+…+(_])"T—+…23n

56sn/(-11„=i«5ஹ(l+x)஺=l+ax+3^~—x2+•••+2!a(a-l)(a-஻+l)஻n!718ᦪarctanxḄ9:;᪍!.=>?/(x)=arctanx,18@U/A)=ߟ^=1—/J"τC|<11+X~MFGH/IJ=/w-/(o)=£]>(,1)“KJn=0L(-1)ᓝ|-ᓰ2஻+1ᵫO/(0)=0ᦑf(x)=£(])"chM2஻+172(06.1)Qᦪ/(x)=—JXḄRᦪ.2+x-%41=/(x)=---------------=^~+^_(2-x)(l+x)2-x1+x2111—13231+x"27Y1K=N(/+>-g"3n=0[J஻=018\=NR+(T)"]x"D«=0LK,173(96.1)8ᦪSᦪ—ḄT.n=22(«-1)=UM஻--1ᑣ

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58m⁚3>7⚪>Q2ᕜḄᦪ!ghᦪ.m⁚4>iᚪXYஹZ[.i᜜P61ஹপஹ(2)ஹ(5)ὃᦻ£ᦪlᑖ᪆¥(¦)ᑢ᳝©ªP96-P602^ᑁᜓ¬᝞asߟ+Z[cicosnx+bsinnx]nn2n=|Ḅᦪ¯ghᦪ.,ஹᦪ.°ᨵ²³ᦪ஻x),g(x)}᝞´f/(x)g(xMx=O,ᑣ¯f(x),g(x)ᙠ[a,ᑗ¶.ᦪ1ஹsinx,cosx,5·2%,(>052%,ߟ-5·஻C,(>05஻CḄឋ>jsinnxdx=0,|cosnxdx=0-zr,¹}jºm=ncosmxcosnxdx,=ீ¼[0tnHn%..[m=nsinmxsinnxdx=<}[0m^njsinnxcosnxdx=0.-nஹᦪḄghᦪ!.{ᳮ7.1᝞´2¹ᕜḄᦪ/(x)ᙠ½`[-º¿¶!⌲⚗ÀᑖḄghᦪ}ᑣᐸᦪ.ao=,Hf(x)dx,a=,H/(x)cosnxdx,n71ᐔ—

59/(x)sinnxdx.஻=1,2,…ÇÈ>QÉ1ÊËÌᙠÍ-º%ζÀᑖᨵ:ᑍ஻n81f(x)dx=—Jdx+Zj[acosnx4-bnsinnx]dx=n-7t2_n=I-nnᑍa஺1y&=—If(x)dx.71'njf(x)coskxdx=—jcosnxcosnx-n2y+hsinnxcosnx]dx=Jacosnxcosnxdxnn-71=ᑍ%jna=—j/(x)cosnxdxn-nÛᳮ=—j/(x)sinnxdx.ᑍ-71FᦪÜ}Ý⌕/,ÉXÊᙠÍ-%,%ζÀ}¶Ḅghᦪßàáᙠ}âQ8஻%)~Ḇ+E[acosnx+bsinnx],tln2=iMæçcos஻x+b“sin஻x]àᔲᦈzOf{x)?{ᳮ2᝞´ᦪ/ÉxÊᙠÍ-ᓛ¹Î¶íî>É1Ê◀ᨵ▲$c,ñ`ò᜜}ᜐᜐ²³ôÉ2Êᑖõᓫ]}÷ᓫ]½`Ḅ$ᦪᨵ▲}ᑣ/ÉxÊḄghᦪᙠ½`Í-º¹Î¶ᜐᜐᦈz}ᐸTᦪ

60f(x),x&(x)Ḅ²³às(x)=)(x+O)+/(x—0)],xù'(x)Ḅcñ`òø—+0)+/(,0)],X=±7l-7T7V~/(F+0)=Um/(x)=-l.Xfᡠ/(x)Ḅᦪᙠ1=஻ᜐᦈ/(+0)+/(())_2[22ᱯ/()ᙠ[-#$%&᜻(ᦪ,ᑣ]an=—1f(x)cosnxdx=0,஻=0,1,2…-nb=[f(x)sinnxdx,"=1,2n71$ᦪ+,/(x)~£>"sinwx,-./0ᦪ.”=1/(X)ᙠHr.2%&Ꮤ(ᦪᑣa=—I/(x)cosnxdx.஻=0,1,2,…n71$d=0,4ᦪf(x)~£a“cos஻x.-50ᦪ./j=0

61628/(x)&2:ᕜ<Ḅ(ᦪᙠ=>#%Ḅ⊤@A.

62J(ᦪL4ᦪ᜻ஹᏔALᣴḄᑁஹ⁚ஹஹ2/ᕜ<Ḅ(ᦪḄ4ᦪ⁚1,%2/ᕜ<Ḅ(ᦪḄ4qᦪrA.⁚2,Msᑭuvwᳮᣴ⁚3,6⚪ᑁஹ=>0,/
%Ḅ(ᦪḄ4ᦪ.⁚1:(1)Ꮤ(2)᜻⁚2,6⚪⁚3,hᚪjkcஹ]ph᜜P691,(1)ஹফஹবஹ2ஹ3£ὃᦻ¦ᦪᑖ᪆©(ª)ᑢ᳝®P124-P127¯°ᑁᜓzஹ2/ᕜ<Ḅ(ᦪḄ4ᦪ2/ᕜ<Ḅ(ovtḄ4ᦪḄqᦪrA&²³´8/(x)2/ᕜ<+ᣚ,x=¶ᐭ/(%).¹71º/(x)=/(ு)=9(y).71ᑣ᜛(y)½&2Gᕜ<Ḅᕜ<(ᦪ¾¿%(p(y+=f—(y+2^)7t=/(—),+2/)=/(—y)=e(y)7171&°(y)ᙠÀ-ÁÂ%Ḅ4ᦪ

63ay(p(y)~-T+Xkcosny+hnsiny]2=]nᐸ¹IL(p(y)cosnydy.14bn=—^(^)sinnydyᐔM71Jy=¶ᐭᨵ:fM~F+£[a“cosᓤx+b„sinÎx]Ï2=1IIW஺஺=",f(x)dxf(x)cos^-xdx.o=7f(x)sin^-xdxwᳮ2᝞Ó/(x)ᙠÀ-/,/Â%ÔÕ:(1)◀ᨵ▲Ø~Ù>Ú᜜ᜐᜐÛÜ(2)ᑖÝᓫßàᓫß=>ḄØᦪᨵ▲ᑣ/(%)Ḅ4ᦪᙠÀ/,/Â%ᜐᜐᦈᐸá(ᦪ:/(X),xᕜXx)ḄÛÜ/(%+0)+/5-0)s(x)=Ú2/H+O]+/(Z-O)x-±l261Jᕜ<1Ḅ(ᦪ/(x)=ãe*,O

64MN26=1,ᦑ6=’,&24=-[f(x)dx=—jf(x)dx=2'dx=e-1lçr/ஹnᐔi1f2/z-zஹ஻Gxa=7Lcos~xdx=~j1)/(X)cosdxnii/-i=2f—excos2n/ixdx=-----------7]21+(2஻]ê=/?஻=ãj=2jexsinInjrxdx2஻G(e-1)1+(2ë2Nᙠ=>0ìÂ%ÔÕsᑭuvᩩîᦑe-18i+(e-l)2^ߟ5(cos2n;vx-2஻sin2n/rx)~rïl+(2஻%)2—e004%Ḅ(ᦪḄ4ᦪ.8/(x)ᙠ0,/%ᨵw©êðñḄ2/ᕜ<Ḅ4ᦪ,◤Jñᑮ%ôᵨḄᨵ,(1)Ꮤ,᝞Óö÷J/(x)ᙠ0,/%Lᡂ50ᦪùº(ᦪ//(x),0

65ᙠÀ0,2Â%Lûü/0ᦪ,Mý/(X)þÿ᜻
ᑣᨵb=£/(x)sindxnr.nᐔxMr2.n^x=[xsin—dx=](2-x)sin^—Jx2஻ττn7rx-----[rxcos——x\-cos----ax]n஻ᐔ2°24nnx,2nnx,fi-nnx,cos12----r1xcos12i—Icos----dx]7rn-2----1n7r2---1--------2Q8,nᐔ(-1)22n-2,k+\=—5—rsin—=n஻-122ᵫ)/(x)ᙠ[0,2],-./(x)ᙠ[-2,2],-(-1)AQk+1)xᦑ=sin---------,xe[0,2]71ᦇ=0(2k+1)2789:;ᦪ/(x)==0