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19971()()2cos,0,0xxxfxax−⎧⎪≠=⎨=⎪⎩0x=a=.12e−.()()0lim0,xfxf→=()2200lncostan1limlim220limcosxxxxxxxxaxeee−→→−−→====21ln,1xyx−=+''0|xy==.32−.()()()()()2'22''22211ln1ln1221,21111211yxxxyxxxyxx=−−+=−−−+−=−−−+''0|xy==32−3()4dxxx=−∫.2arcsin2xC+2arcsin2xC−+()()22arcsin2442dxdxxCxxx−==+−−−∫∫()()()222444arcsin2dxdxdxxxxxxxC==−−⋅−=+∫∫∫5()()()1231,2,1,1,2,0,,0,0,4,5,2tααα=−==−−2t=.3.()123,,2,rααα=12112000452t−⎡⎤⎢⎥⎢⎥⎢⎥−−⎣⎦121200,045t−=−t=3.12111210200042204520452tt−−⎡⎤⎡⎤⎢⎥⎢⎥→−+−⎢⎥⎢⎥⎢⎥⎢⎥−−−−⎣⎦⎣⎦()123,,225rtααα=⇒+=t=3.10x→tanxxee−nxnA1.B2.C3.D4C.0x→()23412!3!xxxexox=++++()()()()()()23tan4tan22333tantan1tan2!3!11tantantan23!xxxxxexoxeexxxxxxox=++++−=−+−+−+()331tan,3xxxox=++tanxxee−()3313xox+tanxxee−3x.3,n=C.()()()tantan000221200201tanlimlimlimsec12sectanlimlim12tanlim1xxxxnnnxxxnnxxnxeeeexxxxxxxxnxnnxxnnx→→→−−→→−→−−−==−⋅==⋅−=−(2)[],ab()()()'''0,0,0fxfxfx()1,baSfxdx=∫()()()()()231,2SfbbaSfafbba=−=+−⎡⎤⎣⎦A123.SSS(B)213.SSS(C)312.SSS(D)231.SSSB.()()()'''0,0,0fxfxfx()yfx=[],ab()(),fxfb()()()()(),.fbfafxfaxaaxbba−+−−()()()()()()()()()()()1213,1.2babbaaSfxdxfbbaSfbfaSfxdxfaxadxbafafbbaS=−=−⎡⎤=+−⎢⎥−⎣⎦=+−=⎡⎤⎣⎦∫∫∫213SSS,B.3()yfx=x()()2'''31,xxfxxfxe−⎡⎤+=−⎣⎦()()'0000fxx=≠A()0fx()fxB()0fx()fxC()()00,xfx()yfx=D()0fx()fx()()00,xfx()yfx=.B()()'0000fxx=≠0x()fx0xx=()()2'''31,xxfxxfxe−⎡⎤+=−⎣⎦()00''001xxefxxe−=()00x≠()''00fx0xx=()fx.(3)()2sinsin,xtxFxetdtπ+=∫()FxA.B.C.D.A.sinsintet2π()22sinsin02sin022sin0sinsincos0cos0.xttxttFxetdtetdtedttedtππππ+===−=+⋅∫∫∫∫A.5()()22,0,0,2,0,0xxxxgxfxxxxx−≤⎧⎧==⎨⎨+−≥⎩⎩()gfx⎡⎤⎣⎦A22,02,0xxxx⎧+⎨−≥⎩B22,02,0xxxx⎧−⎨+≥⎩C22,02,0xxxx⎧−⎨−≥⎩D22,02,0xxxx⎧+⎨+≥⎩D.()gx()()()()()2,02,0fxfxgfxfxfx−≤⎧⎪=⎡⎤⎨⎣⎦+⎪⎩0x()20;fxx=0x≥()0.fxx=−≤()22,02,0xxgfxxx⎧+=⎡⎤⎨⎣⎦+≥⎩22411limsinxxxxxx→−∞+−+++.22411limsintttttt→+∞−−−+−2211141lim111sintttttt→+∞−−−+=−.()22232limsin411xxxxxxxx→−∞−−++−−−222123lim1sin111141xxxxxxxx→−∞−−=⎛⎞++−++⎜⎟⎝⎠(2)()yyx=2arctan25txtytye=⎧⎨−+=⎩.dydx211dxdtt=+2220,tdydyytyedtdt−−+=()221tdyyedtty−=−()()()22121tyetdydxty−+=−arctanxt=tantx=2tan2tan5xyyxe−+=x22tan222tansecsec0,xdyyxyxexdx−⋅−⋅+⋅=()()()2tan21tan11tanxyexdydxyx−+=−3()22tan1xexdx+∫()()22221tan1tan1sec2xxexexxdx=+−+∫()()()()22222222222222222221tan1tansecsec211tan1tantansec2212tan12112tan122tanxxxxxxxxxxxxexexxdxexdxexexexdxexdxexedxexeCexC=+−−=+−+−=+−=+−+=+∫∫∫∫∫()222tan11tan2tansec2tanxxxxx+=++=+2sectanxdxdx=222sec2tanxxexdxexdx=+∫∫2222tan2tan2tantanxxxxexexdxexdxexC=−+=+∫∫4()()2223220xxyydxxxydy+−+−=.,yux=,dyduxudxdx=+()23121uuduxdxu−−=−−231uuCx−−−=221yxyxCx−−−=522123,,xxxxxxxyxeeyxeeyxeee−−=+=+=+−.13xyye−−=2xxyexe−−=21xxyxee−=2xexe−xxe.212xxxyxeCeCe−=++.'212''2122,24.xxxxxxxxyexeCeCeyexeCeCe−−=++−=+++12,CC'''22.xxyyyexe−−=−6111011,001A−⎡⎤⎢⎥=⎢⎥⎢⎥−⎣⎦2AABE−=,E.B0A≠2AABE−=1A−1ABA−−=,1BAA−=−111011,001A−⎡⎤⎢⎥=⎢⎥⎢⎥−⎣⎦1112011,001A−−−⎡⎤⎢⎥=⎢⎥⎢⎥−⎣⎦111112011011001001021000000B−−−⎡⎤⎡⎤⎢⎥⎢⎥=−⎢⎥⎢⎥⎢⎥⎢⎥−−⎣⎦⎣⎦⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦λ1231231232124551xxxxxxxxxλλ+−=⎧⎪−+=⎨⎪+−=−⎩.()()2211154154455λλλλλλ−−=−−=−+−1λ≠45λ≠−.1λ=12312312321,2,4551,xxxxxxxxx+−=⎧⎪−+=⎨⎪+−=−⎩211103331112111211120111455109990000−−−−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥−→−→−−⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−−−−⎣⎦⎣⎦⎣⎦1λ=12311xxkxk=⎧⎪=−+⎨⎪=⎩[()()()123,,1,1,00,1,1TTTxxxk=−+k]45λ=−12312312310455,45510,4551,xxxxxxxxx−−=⎧⎪+−=−⎨⎪+−=−⎩1045510455455104551045510009−−−−⎡⎤⎡⎤⎢⎥⎢⎥−−→−−⎢⎥⎢⎥⎢⎥⎢⎥−−⎣⎦⎣⎦45λ=−.2112112111122103210345516550654009λλλλλλλλλλ−−−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥−→+−→+−⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−−−−+−+⎣⎦⎣⎦⎣⎦45λ=−.1λ≠45λ≠−.1λ=12311xxkxk=⎧⎪=−+⎨⎪=⎩[()()()123,,1,1,00,1,1TTTxxxk=−+k]L()rrθ=(),MrθL()02,0ML0,OMOMLL0,MML.22'20011,22rdrrdθθθθ=+∫∫θ22'2rrr=+'21rrr=±−21drdrrθ=±−21arcsin,1drCrrr=−+−∫1arcsinCrθ−+=±()02r=6Cπ=Lsin1,6rπθ⎛⎞=⎜⎟⎝⎠∓csc,6rπθ⎛⎞=⎜⎟⎝⎠∓32.xy=∓()fx[]0,1()0,1()()'232axfxfxx=+a()yfx=1,0xy==S2()yfx=aSx.0x≠()()'232xfxfxax−=()3,2fxdadxx⎡⎤=⎢⎥⎣⎦()fx0x=()[]23,0,12fxaxCxx=+∈11232003122221122|CaxCxdxaxxaC⎛⎞⎛⎞=+=+⎜⎟⎜⎟⎝⎠⎝⎠=+∫4,Ca=−()()2342fxaxax=+−.()()()211220023421116=3033Vafxdxaxaxdxaaπππ⎡⎤==+−⎢⎥⎣⎦⎛⎞++⎜⎟⎝⎠∫∫()'110153Vaaπ⎛⎞=+=⎜⎟⎝⎠5a=−.()''1015Va=5a=−.()fx()0lim2,xfxx→=()()10,xfxtdtϕ=∫()'xϕ.()()00,00fϕ==,uxt=()()()00xfuduxxxϕ=≠∫()()()()000,0xfuduxxxxxϕϕ⎧⎪=≠=⎨⎪=⎩∫()()()()'020xxfxfuduxxxϕ−=≠∫()()()'02000limlim22xxxfudufxAxxϕ→→===∫()()()()()()'00220000'limlimlimlim022xxxxxxxfxfudufudufxxxxxAAAϕϕ→→→→−==−=−==∫∫()xϕ0x=.ksin2xxkπ−=0,2π⎛⎞⎜⎟⎝⎠.()sin,2fxxxπ=−()fx0,2π⎡⎤⎢⎥⎣⎦.()'1cos0,2fxxπ=−=()fx0,2π⎛⎞⎜⎟⎝⎠02cosxarπ=()00,xx∈()'0fx0,2xxπ⎛⎞∈⎜⎟⎝⎠()'0fx.()fx[]00,x0,2xπ⎡⎤⎢⎥⎣⎦.0x()fx0,2π⎛⎞⎜⎟⎝⎠()000sin2yfxxxπ==−.()002ffπ⎛⎞==⎜⎟⎝⎠0,2π⎛⎞⎜⎟⎝⎠()fx[)0,0y.()0,0ky∉0ky0k≥0,2π⎛⎞⎜⎟⎝⎠0ky=0,2π⎛⎞⎜⎟⎝⎠0;x()0,0ky∈()00,x0,2xπ⎛⎞⎜⎟⎝⎠0,2π⎛⎞⎜⎟⎝⎠.