专升本高等数学-考前课程精讲

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ὃᦪஹ᩽▲ஹ⁚ᦪὃ
1:ᑨᦪᔲᦪᑣ!"Ḅᦪ$"ᦪo1.'ᑡᦪ/(X)-g(x)$"ᦪḄ0()4/(x)=|x|,g(x)=xB.f(x)=x,g(x)=V?c/(6x7-1g(6᯿D./(x)=Inx:g(x)=31nxx+lூBᫀ௃Dூὃ௃ᦪḄE⌕GஹHஹI᪆KூI᪆௃IᑨMᦪ0ᔲ0"ᦪO◤⌕-I᪆K᪵OD⌱⚗I᪆K᪵O0"ᦪ஺A⌱⚗I᪆KV᪵஺ὃ
2ᦪ-,x^0x(1)ᐹYᦪZீ$\,XNOlog]x,x>0arcsinx,arccosx,-1

1
ὃூBᫀ௃(-*-4]U[3,+8)ூὃ௃ὃwᦪḄ஺ூI᪆௃Ix2+x-12>0,(x-3)(x+4)>0,xe(-oo,-4]U[3,+°o)2.xᦪy=/(x)Ḅ$[-2,2],Zᦪ஻2x—4)Ḅ.ூBᫀ௃xe[l,3]ூὃ௃ὃwᦪḄ஺ூI᪆௃I-242x-442,l

2
ὃூI᪆௃I^=2,(6ὃ
4ᦪḄឋឋᓫឋ:ᑭᵨᦪᑨM)(X),Ꮤᦪ᜻Ꮤឋ:-/(X),᜻ᦪᨵឋ,ᕜឋ(Vὃ)1.ᦪ/(x)=1+lnxᙠ(0,e)ᑁ)4ᓫ⌴ᨵᕒᓫ⌴ᨵCᓫ⌴¡Dᓫ⌴¡ூBᫀ௃Cூὃ௃ᦪḄឋ஺ூI᪆௃I/(x)=l+lnxᙠ(O,e)ᑁ0ᓫ⌴ᦪO¡'஺2.ᑨMᦪ¢=ln(x+GTI)Ḅ᜻ᏔឋூBᫀ௃᜻ᦪூὃ௃ᦪḄឋ஺(Jx?+1+%)(>/ª2+1-x)ூI᪆௃I)(-x)=hi¦x+Jx?+l)=lnJ%2+1+X=ln-----=-ln|Vx2+l+xj=-y(x)O¢$᜻ᦪ./3

3
ὃ«⁚᩽▲ὃ
1:ᦪᑡḄ᩽▲᝞®஻¡▲ᜧ°,ᦪᑡ±%,,²¡▲³´µ¶ḄᦪOÏÐaÑÒ·ᦪᑡ±x"²®"f00°Ḅ᩽▲,º»limx”=aᡈ-»a(->00).1.᪷Á⚪ÃÄ(1)ᦪᑡ—……OᐸÈ⚗$23456ூBᫀ௃x„=(-l)n-nூὃ௃ZᦪᑡḄÈ⚗ÌK஺ூI᪆௃IÈ⚗$Í=(-1)"Ln(2)xᦪᑡ1,6…OᑣᦪᑡḄÓ஻⚗S஻=24816limS=.n—>oonூBᫀ௃ÖI᪆ூὃ௃ZᦪᑡḄ᩽▲஺ூI᪆௃I:È⚗$X”,limS“=22/+6஻2-52,ØÙ᩽▲lim3n23+2M+12ூBᫀ௃-34

4
ὃூὃ௃ZᦪᑡḄ᩽▲஺2+72/+6஻252ூI᪆௃I:lim=lim“Too3/+2஻+1”►8"~2r33+«+-?nn0,mnὃ
2$ᦪḄ᩽▲ூ!௃%&᩽▲Ḅ'ᵨ):ᨵ7ÝÞK—,—,00-00,0-00,10°,00°,0°0001.Z'ᑡᦪḄ᩽▲à°,áâOãäâåᨵ:000জçèéêKàë〉ᵨµx-»x0,9;00ঝ◀ï〉®¡ðᜧà〉ᵨµXfoo஻f00°O-å;00ঞᑖôᡈᑖõᨵᳮᓄà〉ᵨµøᨵ᪷ùḄ᩽▲ú⚪åûটÈᑖà〉ᵨµ00-00ঠᑭᵨ᩽▲ÌKà〉ᵨµO,r°ডÿ
ᣚঢ
Ḅឋ
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;ণᑭᵨᜳᳮ〉ত"e#ᡈ"%ᦪ#〉ᵨ'r0,0°(8°5

5,-./0ὃ234থ678#ᑣ(Q,=).0001.BCᑡ᩽▲arctanx-xsinv_xee(1)hm-----z----(2)lim----S—x3)xsin-xூYᫀ௃\]᪆ூὃ2௃B_ᦪḄ᩽▲஺ூ]᪆௃]S_L_iaarctanx-xi+r2..\+21(1)lim---------=hm—=limx=--ioxXTO3X-x->o3-3X-1)sinx-xcosx-11XT஺xsinx3x3xz03x6ὃ23S
*Ḅ▤glima(x)=O,lim(3(x)=0.(1)᝞jlim2=0,kl/no஺p▤Ḅ
rs᜛=o(a),a(2)᝞jlim2=oo,kl᜛noau▤Ḅ
.a(3)᝞jlim2=cwO,klwnoax▤
.ᱯzᙢc=l|,aᓽlim2=l,~|᜛an
rsa(ᨴ.a1.x-0|Cᑡ()nxḄp▤
.ABVl+x-1C.xsin-D.1-cosx6

6,-./0ὃ234ூYᫀ௃Dூὃ2௃ὃ
஺1^2—cosXoூ]᪆௃]Slim—=lim2—=0,ᑨD஺xf°X°X2.x->0|In(cosx)4c”n
ᑣᦪ/=,ᦪ=.ூYᫀ௃A=--,k=2ூὃ2௃ὃ
஺_12%ூ]᪆௃]SlimlnC°fX=lim-05^1=hm^-,A=~-,k^2•soAXx->o4fx->o4x2ὃ24Sᵨ᩽▲]ᦪ⚪r2x2+1ஹ1.lim-------ax-h=2,Bᦪa,bḄ.XT8(X+l/ூYᫀ௃஺=2/=—4ூὃ2௃ὃ
஺2x2+1-(x+1)(ax+¡)ூ]᪆௃]Slim2x2+1-ax2-hx-ax-blim(4x-2ax-b-a)=2,a=2,—a—h=2,b=—47

7,-./0ὃ234
⁚ὃ
1:ᦪḄ,-ឋᑨ_ᦪᙠ¤=2nᔲ¦§⍳©ªC«¬Sজ/(X)ᙠ2®ᜐnᔲᨵ°±Sঝ³´᩽▲nᔲµ¶ᙠঞ³´᩽▲nᔲ¸¹º'_ᦪᙠ»=2Ḅ.1.¼_ᦪ/(8"(1+¾)'¾*°ᙠ¾=0ᜐ¦§Á°஺Ḅa,x=OூYᫀ௃A=--,k=22ூὃ2௃ὃ
஺ூ᪆௃lim/(x)=lim(l+xAÃ/(x)ᙠx=0ᜐ¦§,7ÄÅXTOX->0lim/(x)=/(0),ᦑa=e3.ὃ
2SᦪḄ.
/ᐸ12জÈ=ÉÊ2(¶ᙠ)ËÊ2S/(/-)=/(/+)SÌÍÊ2S/(x-)*/(x+)S00ঝÈÎÉÊ2(/(%-),/(x0+)ÏÐᨵ=ÑÒ¶ᙠ)
Ê2Slim/(x)=ooᡈlim/(x)=oo1.g_ᦪ/(x)=,ᑣx=On/(x)Ḅ..Ê2.W(xSூYᫀ௃ÌÍ8

8,-./0ὃ234ூὃ2௃_ᦪÊ2ÉÔḄᑨ஺ூ]᪆௃]Slimf(x)=lim——ᓃ——r=2,lim/(x)=lim/"——=-2³´᩽7…Ö10-_×2_1)I"'"0-×2—1)▲µ¶ᙠØÒ¸஺x=0nÌÍÊ2஺ὃ
3S3451.ÙÚÛÜÝ᝞j/p/(x)=Z,kàÝá=/âãÝá=/(X)ḄÙÚÛÜÝ.äSᨵ|◤⌕ὃXf+00ᡈXf-00|ḄᓫèÙÚÛÜÝ.2.ᚖàÛÜÝ᝞jlim/(x)=8,kàÝx=/âãÝá=/(x)ḄêÛÜÝ.3.êÛÜÝ᝞jãݶᙠÛÜݺëÒnÙÚÛÜÝìÒnᚖàÛÜÝkâêÛÜÝ.êÛÜÝḄB#஺gàÝy=ax+b(aw0)nãÝy=/(x)ḄÛÜÝᑣᨵlim[f(x)-<2x-Z>]=0ñòóᑮ=a,jim[/(x)-ax]=6XT0°Xy=ax+bâêÛÜÝ.1,Bá=+m(1+j)ḄÛÜÝ஺9

9,-./0ὃ234ூYᫀ௃\]᪆஺ூὃ2௃B_ᦪḄÛÜÝ஺ூ]᪆௃]Slimy=oo,limy=0,y=0âÙÚÛÜÝSXT+OOXf-XJlimv=oo,x=0âùàÛÜÝ10…ú1.lnl+eY/.+]lim—=lim—+hm--------=lim----=1,XT8XX->8XTQOXXT8|_|_^Ylimy-x=lim—+ln———=0+0=0,y=xâêÛÜÝ஺Xf+8'7Xf+oOXIQXJὃ4$ᳮ'ᳮᐵ1.üýþÿx3—4/+1=0ᙠ0,1ᑁᨵ᪷.ூᫀ௃᪆ூὃ௃!ᳮ#$⚪஺ூ᪆௃'(/)=/412+1,/0=1,஻1=2,ᵫ!ᳮ-,ᙠ0,1ᑁᨵ4.//q=0,ᓽ23x3-4᮱+1=஺ᙠ0,1ᑁᨵ᪷.6ᓽ#஺10

10789:;ὃ<=>?@ᐗBᦪDᑖF>⁚HᦪḄᭆKὃ1:ᦪḄᦪஹ"ᦪM஺=⇪Axᡈὅ/'X஺:TᚆV1./Xᙠx=XoᜐYH6ᑣ/A,"/஻/+2ᓝBlin/D/G஺+gD.iim---------------------10Xgo2hC]im/"஺//-2a-X஺…DlimA—஺2AXAs”AXூ$ᫀ௃Cூὃ௃HᦪḄ!b஺ூ'᪆௃'_fxo=lim/x—"x஺,7bᓄn6Coᔠ!b஺X-Xo/'X஺qᙠo£'a஺t£x°vqᙠ6w£%=y'/.ὃ2'ᦪḄ)*+,-᳛ஹᑗ012ᐵ3⚪1.z{y=2x+Inxᙠ1,2ᜐḄᑗ{23.ூ$ᫀ௃y=3x-lூὃ௃BᦪḄᑗ{஺II

11789:;ὃ<=᪆:y'=2+6,=y'(l)=3,y_2=3(x-l),y=3x-Lxὃ3'YHḄᐵBᦪᙠYH6ᑣBᦪᙠ!BᦪᙠᑣBᦪᙠ!YH>?⁚BᦪḄHᑣ;H(1)(C)'=0(஺ᦪ)ফ(yy=஻x஻T,(4y=56(J=-5(3)(axy=axIna(4)(/)'=/(5)(logx\-(6)(Inx)r=—axlnaX(7)(sinx)z=cosx(8)(cosx)r=-sinx(9)(tanx)f=sec2x(10)(cotx/=-csc2x(11)(secx)r=secxtanx(12)(escx)r=-CSCxcotX(arcsinx)r=/1(13)Vl-x2Vl-X2,ஹ61(15)(arctanx)=-----(16)(arccotx)r=-?.l+x1+Xὃ3'¨BᦪH'©Bᦪ'ªᦪ23H1.Bᦪy=2d+4x2—6x+4+7ḄHᦪ12

12789:;ὃ<=ூᫀ௃y'=6x2+Sx-6+-^=2y/xூὃ௃BᦪḄHᦪ஺ூ᪆௃'/=6X2+8X-6+—2«2.(J^=X2+2X-1(X>0),ᑣᐸ©Bᦪx=e(y)ᙠy=2ᜐHᦪ)1111A.-B.——C.D.——4422ூᫀ௃Bூὃ௃©BᦪHᦪ஺ூ᪆௃:1_J_y=2,ᑣ+2x—1=2,x=l,x'(y)=,),¯'Q)=2+2-1x=t-sint,ᑍ3.z{+ஹᙠ6=ᜐḄᑗ{23y=6z(l-cos/)2ூᫀ௃y=axaூὃ௃ªᦪ23°!ḄBᦪH஺ூ᪆௃:dy,dtasint³1y=-----------,y±=>3²ஹ=—Vy=adxI-costdtᑗ{23'y-a-a-ax-—2\a2J13

13789:;ὃ<=ὃ
4$◚ᦪ8':ᦪ8ஹ;ᢣᦪ8=>▤8ᦪ◚BᦪH'23µ¶·¸¹º»¼H'¹ᦪH'23µ¶·¸½¹ᦪ6ᯠn¿HÀᢣBᦪH½eᡈ23µ¶·¸½¹ᦪ6ᯠn¿HÂ▤Hᦪ'Ä▤ஹ?▤ஹÅ▤Hᦪ6ÆÇÈÉÊ.1.23e'+V_3h=0ᡠ°!Ḅ◚Bᦪy=y(x),Ì.(◚BᦪH)ூᫀ௃/=ÍÎ.2y-3xூὃ௃◚BᦪH஺ூ᪆௃'23µ¶·¸¹xH6/e*+2yyf-3y-3Ï’=0,Ì=)——.2y-3x2.Ñ-Bᦪy=(cosx)"s"\Ì.(ÒᢣBᦪH),/isintsinxfl+sinx)x+ூᫀ௃y=(cosx)cosxIncosx--------------------cosxூὃ௃ᢣBᦪH஺ூ᪆௃'23µ¶·¸¹XH6/([l+sinx)]lncosxt(1+sin.tInfcos.riSillX(14-SillX)y=eJ=ecosxIncosx--------------------cosx/ஹi+sinxÓெsinx(l+sinx)=(cosx)cosxIncosx--------------------cosx3.(y=2ஹ6ᑣ/")=.ூᫀ௃y{n}=2x(

142)".ூὃ௃Â▤Hᦪ஺14

15789:;ὃ<=ூ᪆௃:Ì=2vln2,y"=2,(In2)2,…,yಘ=2A(ln2)”.>Å⁚BᦪḄDᑖὃ
1:?ᑖLDᑖ×Ø(1)dC=0Cᦪ;dx,d\[x=-\=dx,d-(2)dx;2yJxx(3)d^ax^=ax

16adx(4)d(ex)=exdx(5)d(log஻x)=—Ú;(6)d[

17x^--dxx

18ad(sinx)=cosxt/x(8)J(cosx)=-sinxt/x^/(tanx)=sec2xdx(10)d(cotx)=-csc2xdx(11)t/(secx)=secxtanx^Zr(12)d(cscx)=-cscxcotx;Ü1—dx"(13)d(arcsinx)=0——dx(14)d(arccosx)=—15darctanx=-^—^dxx(16)d(arccotx)=~~~2^'2.ÝᔠBᦪDᑖDᑖÞ»ឋ(Bᦪà=/,tu=஺gvYH6ᑣÝᔠBᦪy=/᜛gḄDᑖ3dy=*duᡈ2=j"((p(x)W(x)dx.15

19789:;ὃ<=
5⁚6ᳮὃ1:78ᳮஹ9:ᨽ<6ᳮ=>?1.âã!ᳮ(Bᦪy=/(x)äå'(1)ᙠæYç(2)ᙠè(éb)ᑁYH(3)/(a)=/ë)ìí6ᑣqᙠJe(a,b)6.//'য=0.2.ïðᨽò!ᳮ(Bᦪy=äå:পᙠæYçফᙠèôᑁ")-M)YHìí6ᑣqᙠJe(a,b),.//C)=h-a1.õᑡBᦪᙠ÷!äåâã!ᳮᩩùḄᨵ()x+Lx<54/(x)=<,XG[0,5]B.y-I=,xe[0,2]1,x>5VwC.y=xe',xe0,1D.y=x2—l,xe[-l,l]ூᫀ௃Dூὃ௃âã!ᳮḄú#ᩩù஺ூ᪆௃'>=/1ᙠ11,1çäåâã!ᳮḄÅþᩩù஺2.Bᦪy(x)=x67ᙠ0,3çäåâã!ᳮ6ᑣ4=()A.28.3C.OD.1ூᫀ௃Aூὃ௃ᳮḄᩩ஺16

20ὃூ᪆௃/(X)=VT-"#$#(x)=0,%=2.2V3-Xb—a8.&'()*arctanb-arctana<------.2ahூᫀ௃+᪆஺ூὃ௃,-ᨽ/ᳮ0ᐵḄ'⚪஺ூ᪆௃'3(x)=arctanx,ᦑ/'(x)67,-ᨽ/ᳮᩩ#ᙠ(a,b)9:;<ᙠ"=#>?ভ=(B)=—D,b-a1+Jarctanh-arctana1111nᓽn-----------------=——<——<—<——#ᓽnn77b-a1+,1+஺“2a2ab,b-aarctanb-arctana<-----.labKL⁚NᦪḄPᵨὃ1ᓫAឋஹ᩽ஹᨬRᦪᓫTឋḄᑨ&Rᦪy=/(x)ᙠW9XY#ᙠ(Z6)ᑁ\N#]প_ᙠ(a,b)ᑁ//(%)>0,ᑣ.=/(x)ᙠc஻,\9ᓫTefg(a,b)ᑁ\NhiᓫT⌴eklg(2)_ᙠ(a,b)ᑁ/'(X)<0,ᑣm=/(x)ᙠab
9ᓫTn;.(᝛)9ᑁ\NhiᓫT⌴nkl;17

21ὃqRᦪ᩽sḄtuv1wqZNᦪ/'vXw;v2wqz#ᓽ{|/'vxw=0Ḅ᪷#q>/'vxw(<ᙠḄgv3w/'vXwᙠv2wqZḄᓄ#/஻vxw<ᙠ()Ḅz#\ᵫ/஻vxwᑨᔲ᩽s#᩽ᜧvwsgqRᦪᨬsḄtu᝞v1wqz(\NḄgv2wqklஹz(\NᜐḄRᦪs#¡¢sḄᜧ#£¤sᨬᜧ¥ᨬᜧs#£¤sᨬ¥ᨬs.1.ᑡRᦪᙠv0,+8w9ᓫTn;ḄvwA.y-ex-1B.y-4x-3x2C.y-arctanx-xDy=2sinxூᫀ௃Cூὃ௃RᦪNᦪḄPᵨ஺1r2ூ᪆௃y'=——7-1=----7<0,-1+²1+x22.qRᦪ/(³)=/—2/+2Ḅ᩽s.ூᫀ௃+᪆ூὃ௃᩽sḄq´஺ூ᪆௃/,(X)=4X3-4X=4X(X2-1),$/'(X)=0,?z᳝=0,³2=—1,¶=1•/"(x)=12x2-4=4(3x2-1),18

22ὃ=—4<0J஻(—1)=/஻(1)=8>0,ᡠ½ᵫKDᐙᑖᩩÀ#Á=0i᩽ᜧs#᩽ᜧsi஻0)=2gÂ=-1,Ã=1#iĤ᩽s#᩽si/(—1)=/(1)=2.ὃ2ÅÆÇÈklÆÅÇÈkl᪷ÉD▤NᦪᑨgRᦪḄÅ_Rᦪ/(X)ᙠx=x0ËÌD▤Nᦪ<ᙠ]0Í#ᑣ(%,/(%))hiRᦪÎÏ/(X)ḄÅ.Å<ᙠḄÐ⌕ᩩ_(᳝),/(/))hiRᦪÎÏ/(X)ḄÅ#ᑣ/஻(%)=0ᡈ/஻(%)(<ᙠgÅ<ᙠḄᐙᑖᩩ_/(X)ᙠx=5ḄËÔD▤Nᦪ<ᙠ]0Í#ᑣ(%,/(%))iÎÏ/(X)ḄÅ#_0Õ#ᑣ(x°#/(x஺))(Å.1.×ØᑡRᦪÙÚḄÇÈឋ(1)y=ex;(2)y-xy+3x2+5;ூᫀ௃+᪆஺ூὃ௃NᦪḄPᵨ஺ூ᪆௃পy'=2xex2;,/=2(l+2x2)/>0;,ᡠ½^=Ý-ᙠ(-00,+00)ᑁÇḄ.(2)y'=3x2+6x;,y"=6x+6=6(x+l);.ßx>-là#yு0y">0;gßx<-là#y"<0.19

23ὃᵫᳮ#ÎÏᙠv—00,7â9ÈḄ#ᙠc—1,+00w9ÇḄ.2.ãÀv1,3wÎÏy=ax3+bx2y=ax3+hx2Ḅ"¤Å#äqḄsÆÎÏḄÇÈkl.ூᫀ௃+᪆஺ூὃ௃NᦪḄPᵨ஺ூ᪆௃V=3"2+26x,y"=6ax+2b.y'=3ax2+2bx,y"=6ar+26.æv1,3wiÅ#ᦑ67a+b=3,6a+26=0,è?a=—,b==.22y"=9vl—xw,æêᙠv"8,1wᑁ>">0,ÎÏiÇḄgᙠv1,+8wᑁy"<0,ÎÏiÈḄ#ᡠ½v"8,1wiᐸÇkl#v1,+8wiᐸÈkl.20

24ὃKì"ᐗRᦪîᑖï
E⁚FGᑖὃ
1:AᦪஹBCᑖḄ/ᐸឋ1.ðRᦪFf(x)=/(x)2.(îᑖḄឋóপ(]7*ö)=/'(x),“/(xL=஻x)ᡃফJf:(x)dx=/(x)+C,j;x)=/(x)+C;(3)|[/(x)±g(x)]tZr=j/(x)cZr±jg(x)d!r;4ᝯ'=%//ᦪ3.ᑖ"#$%পJ\Odx=C(Cᦪ)*ফJxkdx=---xk+]+C*k+\বj—t/x=ln|x|+C(4)[axdx=—ax+CJ

25aJexdx=ex+CJsinxdx--cosx+Cম(6)Jcosxdx=sinx+Cjsec2xdx=tanx+C9(8)>Jtanxdx?@In|cosx|+C(10)Jcotxdx=In|sinx|+C(11)JcscxtZx=ln|cscx-cotx|+C(12)Jseexdx=In|secx+tanx|+C(13)jesc2xdx=-cotx+C(14)Jcscxcotx4/r=-cscx+Cf/1d^=arcsinx+CJsecxtanxdx=secx+C(15)(16)JVl-x221

26HIJK#ὃMNOr]1x(17)[-----dx=arctanx+C(18)-----dx=-arctan—+CJ1+xJx+aaa/ஹS1,1ix-aS(20)f/1dx=arcsin—+C(19)I-----rt/x=—In------FCJx-a2aX+QJyla2-x2a1.YZ[ᦪ/(x)\][ᦪ^_`(x)/(x)Ḅ@bc[ᦪ^ᑣeᑡᐵh%ᡂjḄk()4lJ/(x)dx]=/(x)dxB.(j/(x)t/x),=/(x)C,jF(x>/x=F(x)+C£>.j/(x)c/x=F(x)+Cூqᫀ௃DூὃM௃ᑖḄឋv஺ூx᪆௃xzJ/'(xMx=/(x)+C2.|/(X)Ḅ][ᦪkCOSX,ᑣ/(X)ᨵ@bc[ᦪ()A.1+sinxB.\-sinxC.1+cosxD.1-cosxூqᫀ௃DூὃM௃][ᦪc[ᦪḄᐵh஺ூx᪆௃xz/(x)=sinxcosxḄ@bc[ᦪ,Jf(x)=-cosx+C.C=1dx.ூqᫀ௃x᪆ூὃM௃ᑖ஺ூx᪆௃xzc%=J(஺'$ߟ=jT+஺22

27HIJK#ὃMNOὃ2$FGᑖḄ
EᣚᐗKLMᑖK᪶z^Ḅ⌕ᙠ[ᦪᑮex"X,r^'-l,1.I--------dxJex-\ூqᫀ௃x᪆஺ூὃM௃@ᣚᐗᑖ஺ூx᪆௃xz_2x1+1^dx=e'+x+C2.tan3xdxூqᫀ௃x᪆஺ூὃM௃@ᣚᐗᑖ஺ூx᪆௃xz[tan3xdx==1(sec2x-1)tanxdx."=secx,22-InM+22iIIMk@Isecx—In|w|+-^-+C=—In|secx|H----——FC.ூqᫀ௃x᪆஺ூὃM௃@ᣚᐗᑖ஺ூx᪆௃xz23

28HIJK#ὃMNOjsin4xdx=Jl-cos2x2“x=M(1-2cos2x+cos2xVx2*J(l-2cos2x+cos22xylx=^-(x-sin2x)+|cos22xdx%-sin-ஹrl+cos4x,1/.-ஹ1sin4xS2x)4-J——ox=~(x-sin2x)+—x+\-C4Vὃ
3$BCᑖḄDEᣚᐗ⌕kᣚᐗzcos2x=l-sin2x,sec2x-tan2x=l*᪷%ᣚᐗXூqᫀ௃x᪆஺ூὃM௃?ᣚᐗᑖ஺ூ᪆௃xzA,...fV4-X2rV4-4COS2/.x=2cost,dx=-2sintdt,-----*—dx—-------—atJxJ4cost=fsM,>=-f——^—dcost=--——+C=——------FCJ2cos"tJ2cost2cos/2cos'ஹ2ூqᫀ௃x᪆஺ூὃM௃?ᣚᐗᑖ஺ூx᪆௃xz.rJ/—9x=3sec/"=3sec/tan(^-------dxJx=3j-^-^-secttantdt-3jtan2tdt=3j(sec2/-1)dt=3tant-3t+Csect24

29HIJK#ὃMNO=Vx2-9-3arccos@■\-Cx3J¡$ூqᫀ௃x᪆஺ூὃM௃?ᣚᐗᑖ஺ூx᪆௃xzx=2tandx=2sec2tdt,/τOt/x=j~sec_j?>-In|sec/+tan/|+Csecyj4+x?2sect+C4.[j+3dxJJ2x+1ூqᫀ௃x᪆஺ூὃM௃?ᣚᐗᑖ஺ூx᪆௃xzlS2x+61r2x+l+5.1r2x+l.1r5,=-I-/:=-I—,-dx=-I-?dxH—I-.dx2JJ2x+12JJ2x+12JJ2x+12J¢+1iHj-------5P-1135i=—jy/2x+\dx-\--J2x4-12dx=—2x+12+—2x+124-CὃM4zᑖḄᑖ£ᑖ¤ஹஹ¥ஹᢣஹ*⌕§¨©25

30HIJK#ὃMNOª஻="(X),V=v(x)k\[ᦪ^_u'(x)-v(x)ᡈu(x)-v'(x)ᨵc[ᦪ^ᑣᨵᑖ£ᑖ$%:J"(x)•v\x)dx="(x)•v(x)-jv(x)•஻'(x)dxᡈudv=uv-[vdu.1.jxe~'dxூqᫀ௃x᪆஺ூὃM௃ᑖ£ᑖᑖ஺ூx᪆௃xzJxexdx--jxde~x-@1""@je~xdx)=-xe^x-e~x+C2.jexsinxdxூqᫀ௃x᪆஺ூὃM௃ᑖ£ᑖᑖ஺[»Wzlf^=jsinxJev=eAsinx-jexcosxdx=e"sinx-Jcosxdex=exsinx-e*cosx+jexdcosx-ex(sinx-cosx)-jexsinxdx.ᡠ¶^jexsinxdx=-^ex(sinx-cosx)+C.?⁚ᑖὃM1zᑖḄ¸1.ᑖḄ¹º»¼½¾ᐵ()4ᑖ¿ÀB.[ᦪCᑖÁ▲஺.ᑖe▲ூqᫀ௃A26

31HIJK#ὃMNOூὃM௃ᑖḄ¸஺ூx᪆௃xzᑖḄᦪ¹ᑖ¿À¾ᐵ஺2.YZ/(X)ᙠÃÄ஺,\Á\^ᑣÅÄ/(=.ூqᫀ௃/(X)ூὃM௃ᑖḄឋv஺ூx᪆௃xz*Bf(x)dx=/(X)ὃ2zGᑖḄ)*+(1)ª/(x)>0,ᑣÄ/(xRxḄ¹ÇÈ/(x),x=a,x=6,xÉᡠÊᡂËÌÍÎḄ☢.(2)ªᑣJ/(x*xḄ¹ÇÈ/(x),x=a,x=6,xÉᡠÊᡂËÌÍÎḄ☢ḄФᦪ.(3)ª/(x)ᨵÑᨵÒ^ᑣḄ¹ÇÈ/(x),x=a,x=b,xÉᡠÊᡂÔÎÕÈxÉÁ¨☢ÕÈxÉe¨☢Ö.1.ᑭᵨᑖḄÙÚÛ¸ᑖz(1)JJx|tZx*(2)£y/a2—x2dx(tz>0)*ூqᫀ௃(1)3(2)-Tia14ூὃM௃ᑖḄÙÚÛ¸஺ூx᪆௃xzÝÞßÔàá☢ᓽ\஺ὃ3zGᑖPᙠḄᐙᑖᩩT1.eᑡãÑäḄk()27

32HIJK#ὃMNO4|“X)ᙠÄæçèé^ᑣêLìᙠ*A|/(x)ᙠ^\]^ᑣJz/(x)$@ìᙠ*C|Ä/í)$ìᙠ^ᑣ/(x)ᙠÄæçÁ@èé*஺.|£'/(x)$ìᙠ^ᑣ/(x)ᙠÄ“,6Á@\].ூqᫀ௃Aூὃ௃ᑖḄឋv஺ூx᪆௃xz[ᦪᙠÃÁèé^ᑣᑖ@kìᙠḄ஺ὃ4$GᑖḄឋUª/(x),g(x)\,ᑖᐹᨵ¶e⌕ឋvzឋv1JÄ/(x)±g(x)W=jf(x)t/x±jg(x)dx..ᓽ[ᦪïឋðáḄᑖÇÈñòḄᑖḄïឋðá.óôõbឋvö〉ᵨÈᨵ▲øb[ᦪïឋðáḄùÎ.ឋv2Jz0(x)dx=4z/(x)dx.(úᦪ).ᓽ[ᦪḄᦪ¼û\¶üᑮᑖýþ☢.ឋv3¾ÿa,b,cḄ᝞ឤᨵJ/(x)=£/(x)+J/(x)dx.⊤ᑖᐹᨵ"#$%ឋ.1.)ᑖJூ+ᫀ௃./᪆ூὃ௃ᑖḄឋ1஺3/᪆345=J+g+2=g.28

3389:;<ὃ>?2.)ᑖJ஺Jl+cos2xdx.ூ+ᫀ௃./᪆ூὃ௃ᑖḄឋ1஺ூ/᪆௃45______7Z-£yjlcos2xdx-D/|cosx|dx=^2/cosxdx+JF(-cosx)dx'~L2_(A£-\B2sinx-sinx=y/2(1-0-0+1)=2^ឋ14Nᙠ"#3a,$P/(x)NO,ᑣJ/(x)20.TU1Nᙠ"#3a,bVP,஻x)ீg(x),ᑣ//(x"xீZg(x)dx.TU2j/(x)tZx

3489:;<ὃ>?ூ+ᫀ௃./᪆ூὃ௃nP▲ᑖbᦪ)஺ூ/᪆௃y᩽▲“9”Ḅ5ᑭᵨᑣnP▲ᑖbᦪoᦪ512J஺(l-cos/)d/—X..1-cosx21lim•=lim-------=lim=I.XTOx33x*3஺3x6ὃ6GᑖḄXY1.-¡¢₈5abᦪ/(x)ᙠ3z$Pde/(x)h/(x)ᙠ3a,¤PḄir4bᦪᑣ|V(x)dr=F(x)|*=F(b)-F(a).2.ᣚᐗᑖ᝞/(x)h3a,$PḄdebᦪx=©ª«ᑡᩩ®প(p(a)=a,(p((3)=b;ফx=e(f)ᙠ3tz,°(ᡈ▯aV)ᨵdeᦪ.(f)Ḅu³(c3a,Z´V,ᑣJ/(xg=J/3᜛(/)¶(/)F.3.ᑖ·ᑖjudv=uv^-Jvdu.a1.¸¹ᑖ/1º+3»ூ+ᫀ௃-530

3589:;<ὃ>?ூὃ௃ᑖḄ¸¹஺ூ/᪆௃/¼»++»=½:i;31:=()rx+22.dxJ2x+1ூ+ᫀ௃—3ூὃ௃ᑖḄ¸¹஺ூ/᪆௃/À2x+l=t,x==tdt,x=0,/=l,x=4,/=3,ᑣ4ᑖ,-(Z2-l)+24x+2,0~~/•dx-tdt=2+Á=+|Â2x+lὃ7$Ꮤ[ᦪ\᜻[ᦪᙠ^_`abḄGᑖabᦪ/(x)ᙠ[-.$Pdeᑣ“(0J(x)h᜻bᦪÅ[f(x)dx=pa120/Æ.,/5)hᏔbᦪ1.jx3eᵨdx=ூ+ᫀ௃0ூὃ௃ᑖḄ¸¹஺ூ/᪆௃1*1h᜻bᦪ᜻bᦪᙠ}"#PḄᑖh0.31

3689:;<ὃ>?
⁚cGᑖdGᑖeᵨὃ1:cGᑖḄᭆhdXY¼f{x}dx=lim\hf[x}dx',Jaa|f(x)dx=lim[j\x)dx\J-coo-»-ooJaf-t-xf0p+coff(x)dx=\f(x)dx+\f{x}dx.J-COJ-QOJOr+oo21.)J஺xe~xdx.ூ+ᫀ௃-2ூὃ௃8{ᑖḄ¸¹஺ூ/᪆௃45=limxe~xdx=--lim\he~xd(-x2]=--lim(e*-1)=-.bT+aoJ஺2bsJ஺\'26-1/2ὃ2GᑖḄeᵨ)☢bᦪÊ=/(x)ᙠ"#[D¤PḄᑖḄᦪuËpᵫbᦪy=/(x),x=a,x=bxÍᡠÏᡂḄÑ☢"³஺Ḅ☢ḄÒᦪ.)ÓÔÕÕ1.ᵫdeÖ×Ê=/(x)vØ×Ù=஺,X=F(஺Û6)ÙÍÏᡂḄÑ☢ÜÝÞxÍÓÔiᕜᡠᡂḄÓÔÕḄÕhᓃ=%J/2(xWx.2.ᵫdeÖ×x=/(y)vØ×Ê=஺/=1(஺Û1)ã»ÍÏᡂḄÑ☢ÜÝÞyÍÓÔiᕜᡠᡂḄÓÔÕḄÕh/=äg2(y9.1.)ᵫÖ×y=/vy=2x—/ᡠÏÜÝḄ☢.32

3789:;<ὃ>?ூ+ᫀ௃-3ூὃ௃ᑖḄDᵨ஺ூ/᪆௃/ᐜæçᡠÏḄÜÝ(᝞«Üᡠè)\y=x2ᵫéêëìᩩÖ×Ḅíh஺(0,0),/(1,1),sxh[y=2x-x2ᑖnîï£[0,1].ᵫ5:A=J(2x-x2-x2)dx=[x2-yX3]o=.()x2V22.)ðᙊr+i=1ᡠÏᡂḄÜÝḄ☢.a~b~4,ூ+ᫀ௃-7cab2.3ூὃ௃ᑖḄDᵨ஺ூ/᪆௃/:஺ÞxÍÓÔòÓÔÕ$óôPõðᙊ^ö^^÷^i^^^^^vøÍᡠÏᡂḄÜÝÞïÍÓÔiᕜᡠᡂḄ.a22xy>ðᙊ—y+^-z-1ᡠÏᡂḄÜÝÞxÍÓÔiᕜᡠᡂḄÓÔÕḄÕa2b2h:_________2a2-x2)dx-2ᐔb213\\a202-xyix=-7rab.333

3889:;<ὃ>?úûüýᐗbᦪþᑖÿὃ
1:HᐗᦪḄஹ᪆ᐗᦪZ=/(x,y)ḄḄ☢ᐗᦪz=/(x,y)Ḅ஺!☢ᙠX஺$☢%ᢗ'(.1.*ᦪz=ln(y-0-Ḅ.A/4-X-yூ1ᫀ௃Bz\y>x>0,x2+y2<4}ூὃ5௃6ᐗᦪḄ஺ூ7᪆௃ᦪḄᓽ:;ᦪᨵ=ᡠ?ᦪḄ@|z|_y>x>0,x2+y2<4j.2.I/(x+y,h)=X2+3L+ஹ2+5,*ூ1ᫀ௃f(x,y)=x2+y+5.ூὃ5௃6ᐗᦪ*7᪆O஺ூ7᪆௃7@/(x+y,xy)=x2+3xy+y2+5=(x+y)'+Ay+5=M2+v+5,/(x,y)=x2+y+5.ὃ
2@HᐗᦪḄ᩽▲lim/(x,y)=Nᡈlimf(x,y)=AX»0(Xy)T(Xo,Z)y-^yo[\@]ᓄᡂᐗᦪ*᩽▲஺X2'11x+y1.bclim1+—.(awOhᦪ)iIxy)34

39klmnoὃ5pqூ1ᫀ௃mூὃ5௃6ᐗᦪḄ᩽▲஺xy(x+y)ூ7᪆௃7:rO=lim1XTOOy—>aஹ\xyslim11+—t=xylimfl+-X-»oOxyjf—>8x21='.rO=/vlim=limXT8xy(x+y)XT8av->ay->a4i+xὃ
3I8ᦪḄஹᐰ?ᑖ1.xyᦪḄ:/(X஺+AX/O)/(XO/MJfz(xo/o)=[iᔩAx/(xo/o+Ay)—/(xo/o)/z(Xo/o)=lim2.}▤xyᦪ35

40klmnoὃ5pq3.ᐰᑖ.dz.dz.az=—ax-\--aydxdy1.*ᦪZ=x2y+2q?Ḅxyᦪ..dxdyூ1ᫀ௃7᪆஺ூὃ5௃6ᐗᦪ*xy஺ூ7᪆௃7@—=2xy+2yi,—=x2+4xy2.dxdyὃ
4@KᔠᦪḄ8ᦪᔠᦪᑖ\O\ᑣ1.Iᦪz=u=,v=dzdzdudzdvdzdzdudzdv——-------1-------l—=--------1-------dxdudxdvdxdydudydvdy2.I஻=/(஻#)ᨵḄ▤xyᦪᦪ஻=஻(%)#=y(x),ᑣdu_dfdu_^dfdvdxdudxdvdxὃ
5@◚ᦪḄ8◚ᦪᑖ\Iᦪ(x,y)=Oy=/(x),ᑣᒹ=—.dxFyIᦪ(x,y,z)z=f(xMᑣ==ߟdxF_dyF.1.Iᐗᦪz=z(x,y)ᵫ[lnW=OᡠḄ◚ᦪ*zy36

41klmnoὃ5pqdzd2zdx:dx2'ூ1ᫀ௃7᪆஺ூὃ5௃ᔠᦪḄyᦪ஺ூ7᪆௃7@tF1x,y,z3=--ln-,F'x,y,z=-,F!==xzyzzzzὡ'1x,y,z3#O,ᓽzwOᵫ=—x*xyஹ,dxF"x+zᦪ¡=ᑮz=z(x,y)Ḅᐗᦪ,5z/x,dzy•—(x+z)—z1+--29;•dz_/Idx)_z~5x2(x+z)2(x+z)337

42klmnoὃ5pq¤¥¦§¨ᑖὃ
1:EMCᑖḄNOPஹឋ©ª=@§¨ᑖJJ7(x/)dcr°ᦪ±/(x,y)NOᐸᦪD±³´?(஺µ?☢z=/(x,y)⚔Ḅ⚔·¸Ḅ¸¨z/(x,y)KOᐸᦪ±³´º»⚔·¸¸¨Ḅ¼½ᦪ.ឋ¿@ឋ¿1(Àឋឋ)Jj[A|/(xM±&g(xy)]dk%[JJ/(x,y)db±&JJg(x,y)db.(4hᦪ).DDDឋ¿2((ÇÈឋ)᝞Ê(஺ËA஺2ÌÍᐳᑁÐÑ5ᑣJJ7(x,y)dcr=jj/(x,y)dcr+JJf(x,y)da.DD2ឋ¿3(ÒÓឋ)ᙠ(஺%Ô஻x,y)20,ᑣf(x,y)Acr

43klmnoὃ5pqឋ¿7⧨ᑖã±ᳮäIᦪ/âxjäᙠÞ(஺%b஺Ḅ☢¨ᑣᙠ(஺%åæçᙠ5âg,஻ä:èOᡂéjj/âx,yädb=/âJ,z7äb.Dឋ¿8⧨ᑖḄ᜻Ꮤìឋäíî´ᐗᦪḄ¨ᑖ(ᐹᨵìឋ§¨ᑖḄ¨ᑖ(ðᨵìឋñòᡃôõö§¨ᑖḄ᜻Ꮤìឋ.1.I(஺/+/44,ᑣ§¨ᑖJJ24஺=.Dூ1ᫀ௃84.ூὃ5௃§¨ᑖḄ©ª=஺ூ7᪆௃7@ᙊḄ☢¨ø¨ᦪ2,§¨ᑖḄᦪ±³´ᙊḄ☢¨Ḅ2ùᓽ2^-22=8^.ὃ
2@QRᙶ᪗UVḄEMCᑖ1.ûüᙶ᪗ÿ
ᑖᙶ᪗Ḅ☢ᐗdcr=ᓽJJ/'xMdcr=.DDজᐜy#xX%&ᑖ'(஺*⊤,-a

446789:ὃ<=>\\f{x,y)dxdy=1do(xj)dy.Dঝᐜx#EF%&ᑖ'(஺*⊤,-஺GHGI%3&«%ீ஻23&0/ᓃH&ᙠ'(஺2Ḅᑖ3ᙠᑣᨵ0/x,ydxdy=J-dyj---/x,dx,21.T஺UᵫWx=2,y=xXY=1ᡠ[ᡂḄ]☢'(^JfJ_x4%Dy9ூbᫀ௃-4ூὃ<௃ᙶ᪗
ᑖ஺ூf᪆௃f-ᑖ'(᝞iᡠjὶlx=2,y=xXm=1ᡠfno<ᙶ᪗ᓽ/(l,l),8(2,£j,C(2,2).22ᡠtu=J-(T+/"x94Dyxyூyz௃ᙠᙶ᪗{|ᙢᑖḄ
~:জᑖ'(ᑨ'(%;ঝᑖᑖᓄ,ᑖ;40

456789:ὃ<=>ঞ
ᑖnz.2.᩽ᙶ᪗
ᑖ=jj/rcosO,rsmO^rdrdODD'᪷ᙶ᪗mDḄ_:1&᩽drD,°3&᩽

466789:ὃ<=>ᑣD'={(r,0)|O<0<2^,O

476789:ὃ<=>ঝᑖÚÛḄ2▲ᑖᓄ,ᑖ;ঞ
ᑖnz.43

486789:ὃ<=>ÜÝÞßàᑖÃÄὃ
1:WᑖXY*)Z᝞á=â=/(x)g(y),*ᓄ,Iä=/(x"*¯æçᑖn-dxg[y)J—J/(éÎ+஺nêf.ὃ
2-[\?ᑖ)Z᝞=ë(2],*í஻=—,y=ux,--u+x—,îᐭÃÄmdxJxdxdx஻+éð=ộ/),òóᨵô=ᙠ,¯ᑖn1X=õ,dx(p(u)-uxJ(p(ii)-uJx^ᑖ#öᵨ2îø஻nᑮúÃÄêf.XJfdy^\f[x}dx+C,nêf.ὃ
3-▤5ឋ)Z᝞û+P(x)y=Q(x),êf,-dx1.^ÃÄü'+y=cosxḄêf.ூbᫀ௃y=-(sinx+C)Xூὃ<௃^f{▤WឋÃÄ஺Iccqrூf᪆௃f-ÃÄ*ᓄ,"+2^ÿ
▤ᑖ
XX44

49ὃḄy=eJ"'+C=-(sinx+C)JXJXὃ
4E▤'Uᦪ[\?ᑖ)Z,᝞B+P(x)@+0(x)y=O34(x),Q(x)7ᦪ
9:;▤7<ᦪdxdx=>ᑖ
?yn+py'+(y=O.J/+pr+q=oᐸᱯL
ᑣ᪷OᱯLPQ᪷ḄRST,
ᢥ᯿Wᑡ⊤BᑏDᑖḄᱯLr2+pr+q=0ḄPQᑖy"+pp+Ṻ=0Ḅ᪷᜜
YPQR[\Ḅ]᪷Y
Yy=Ce/]X+Cer2Xl2PQ[\Ḅ]᪷ᧅY0y=(G+஺2ஹ)*_ᐳab᪷c2=a±ig1y=eax(C,cos/3x+Csin/3x)22.oᑖ_/+2/+5y=0Ḅ.ூsᫀ௃y=e~xcos2x+Csin2x).2ூὃ௃o;▤7<ᦪ=>ᑖ஺ூ᪆௃ᱯL/+24+5=0,=-\+2i,r=-1-2/,2ᡠxy=e-*(GCOS2X+Csin2x).245

50ὃ{|}~ᦪὃ
1:]^_ᦪḄᭆa1.ᦪᦈᐙ⌕ᩩᑖᙠ᩽
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