《微積分》各章習(xí)題及詳細(xì)答案

經(jīng)典word整理文檔，僅參考，雙擊此處可刪除頁眉頁腳。本資料屬于網(wǎng)絡(luò)整理，如有侵權(quán)，請聯(lián)系刪除，謝謝！第一章函數(shù)極限與持續(xù)【2】x(sin)1cosxfx).f則2(43x)2.xx)2x0tanxsinxx是.x,1limxsin0k.kxx0limearctanx.xx0exx(x)0則b.f在xx,x0ln(3x6xlim.x0(x)[fx)設(shè)f則1ln(x2)yxaxa設(shè)a則lim()________.xx10)1cosx1ax,ax2與.33x(x)arcsinf1x(x2x22.xx2axalim()8a則設(shè)xxlim(nn1)(n2n)n(xg(x)[l,l]h(x)[l,l]是.設(shè)ff是,(x)g(x)f(x)h(x)f(x)[g(x)h(x)]）f(x)g(x)h(x).）1x(x)(x)1x3x1.,1x;;）與;）~.1x1,x0(xf(x)0則k.在x1x13kx032;;）1;）0.23lim[ln(nlnn].n（Ａ）1;1;）.sinxxx0x0x0x(x)0x0f(x)則是.f1xcosx.(x)g(x)和）f(x)lgxg(x)2lgxf(x)x(),gxx2;f）f2,;(x)xx()3,gx1f(x)1()sectan,gxx.x3x）x3422x=（）|x|x0.1x)（）xx0;e1.e(x)在xlimf(x)）f0xx0.x(x1x)2（）x1）;D2a},,{c}limalimblimc）設(shè)且nnnnnnnnnbnbcn）a;）n;nnnlimac;limbc.nnnnnnx211當(dāng)x1e）x1x1;.xcscxcotxlim2sin）lim）））;n2n1xnx02x13x2x11limx(elim;）;xxx8cosx2cosx11xxx2lim）;;2cosxcosx1xx32xx0111x2)3）;）lim.1223n(nnarctan4xx23211x2,blima使axb.x12x1111123nn1lim.111123nna0xax(n)limx.1且n1nnnnnxxx(x)f.nnxn(x)[a,b]在af(x)b(a,b)f).設(shè)f且內(nèi)至少有一點使xxx2sin2xf(sin)12sin)22sin.22,222f(x)22xf(cosx)22cosx2sinx222.(43x)29xx20.0.xx)xx23xxtanxsinxtanxcosx)limlimcosx)0,xxx0x0x0tanxsinxx是.k0.11sinlimxsin0limx0k0即.kkxxx0x00limarctan0limearctanx(,))..exxx22xx2lim()lim(fx.xbb)lim()lim(fx2,b,exx0x0x0x0f(0)b,b2.1ln(3x6x3x1limlim.26x2x0x08.1xe0lnx11xe.e2y1ln(x(yln(xx2eyx1,y1,xe2y1ln(x2)ye2.y1,x12axaxe2=)2ae2a.a2axaxax3111)1~(1x)1axcosx1~xa由22（運用教材a）與2,以及323211ax2ax)12233limlima1,1cosx13x0x0x223a.211x423x111111x1x42,f(x)x.42101xx(x2x2)(x2x2)22220limx2x2lim22x2x222xxx2(x2)22lim0.x2x222xx2axaaxaa2lim()lim(1)ata令xxxaxxx2a11lim()lim[(1)](1)e83=3xtaaaxattxt1ln233aln8aln8ln2.3(nn1)2(n2n)nlim(nn1)(n2n)limn11)n2.2n11n選（Ｄ）令F(x)f(x)g(x)h(x)f(xg(x)[l,l]是h(x)[l,l]是,由上的偶函數(shù),上的奇函F(x)f(x)g(x)h(x)f(x)g(x)h(x)F(x).()1x1xx()xx)x1x)]3x3x1x1x11x32lim(1x)1ax）a1x1x)x)31x1x13221lim()limlim(1)1xax））fxa1x13x0x0x0x31lim[ln(n1)lnn]limln(1)1nnnn(0)1f(0)0f(0)0f,,f(x)lnxx0g(x)2lnx,而的界說域為選（Ｃ）在（A）中2的界說域為x0f(x)g(x)(,)g(x)xx0f(x)x,2f(x)1g(x)secxtanx2{xR,x}f(x)g(x),2x|x|xx|x|x11,xxx0x0x0x0x|x|x011)((xxe1,xxx0x0limf(x)xf(x),,而9.,0xx011f(x)在xlimf(x),limsin,1sin1有界,0xxxx0x0x0(x1x)(x1x)x22limx(x1x)limxlim（2x1xx1x22xxx1121x11x2）當(dāng)n不n.·.x2111elim(xe200x1x1）x1xxx2111x當(dāng)xe(xex1x1x11x.xxlim2sinlim22x.nn2n12n1nn1cosxx2cscxcotx1cosxxsinx12sinxsinx2limlimlimlim.xxx2x0x0x0x011x(ex1.xxxx2x12x12111lim()))x3.3x3x22122x1xxxx111[lim(1)x1][lim(1)]e333221212xxxx8cosx2cosx1(2cosxcosx(2cosxx2limlimx32cosxx12x31414cosx1cosx12lim2.12x311xsinxcosxxtanx1xsinxcosxlimlimxtanx(1xsinxcosx)x0x0xsinx1cosx2x2xsinx2x21cosx113limlimlim.2x2244x0x0x0lim(1xsinxcosx)2x0111]1223n(nx11111)()()]223nn1x1)1.n1x2x)2x11133limlimlim()33.2xx24arctan4x4x2x232x23x21x212(ab)xbx1(b)x1xxa)x(ab)xb)12x12x1a0a1123(ab)b211111123nn11111n112n1111112311nn1lim11lim1.而1n1xx123nn1,xaaa21kxakaa設(shè)n,,則xax2k1k數(shù)列{x},n再證{x},nxa1且0xn1nxxxnnnnxx{x}limxlimxA即設(shè),n1nnnnnnaA0limxaAAAa,nn1x0x0nn2x12x1f(x)lim0得n1x0limf(x)1limf(x)1f(0)0而x0x0f(x)((0,)x0F(x)f(x)x,則F(x)[a,b]在而F(a)f(a)a0Fb)fb)b0(a,b)F)0,使)0f).即ffh)f2h2limf則h0f(x)(0)(0)0limf有f則xx01arctanyxyf則xx1(x)yfx)yy則,exe)y.xyy則dydydxdxsinx24則21t)limt)ft)若f2則txxxx1y2xey(0).設(shè)y則xdyy(x)ecos(xy)0xy________y則.dx1xt2dy2.設(shè)則ycost21yx2則tan=（.y和x1123.;）(x)ef()ekftank且則（.x41112;）.2fx)f2x(x)且lim2yf(x)(在.fx04x6y4x2）yx3yx1.yf2(xx)fx2(x)(x)設(shè)f則x02f(x)2f(x)2f(x)f(x).0;）f(x)[f(x)]2則f(x)(x)f且n()[f(x)]n1f(x)]n1）(n[f(x)]n1(nf(x)]2.f(x2x)f(x)(x)xlim若f則（）020xx02xx4x4;x.;;）000(x)()()()()()fxfx）fxfx和fx則fx000000;.(x)x(xx2)(x99)f(0)則設(shè)f（）9999;）;.u)yf(x)（）若f且2()(x)2()2()2()xfx2dxxfx2dx）fx2dxxfx2dx.f'(0)00f且））(x)(0,)在f(x)(;）f;在)f(x)f(0)x(,0)f(x)f(0)x有有.12sinx0x(x)0設(shè)f在x）xb0xb0b0ab）a）a;））;;ab.xlntsin1d2ye）y求;）求;xyt32t1d2yarctanyyyxx求）x）y,;）y(50);2x()y求;x1x(x)x(xx2)(xf(0)）f）f求;(x)(xa(x)(x)xaf(af(a),在求;d(x)x1在f2limf(cosx1).f且求xdxsinx)a2x0b,bf(x)a.eax1x0yaba,b.x22與（問.(x),xx有fxx()()()f(0)1f(x)f(x)fxfx且.fy121212x3x5(.32fh)f2hfh)fh111limlim()f122h0h0f(x)f(x)f(0)x0(0)limlimf(0)fxx0x0lnln|lnxxxy1yxx1)sin)cossin)sinx2xfxxfxx,f)yfsin)cossin)sinxx,yfx2xfxxe110(ln(eeke1y(e)ee1xln(exln(eye1當(dāng),.xxx)x)]2111dxdx)x)x)1x)2x)x)]2dy4x2x2x2x2sinxcosx4x4xsin2x34,2444334dxdydy2xsin2x42dx22xdx1ft)limt)tef(t)etee2t2t2tx2t2t2txxy2x221y112由xx0,200yx1222yexe,yeexexxxxxy(0)ee200eysin(xy)xyxyyxy')sin()(')0exyxyexsin(xy)xyeyxy)xyy'.exxy)xysinttcosty't,ttx't3txd2yd(y')dx(y')'1tcostsint1sinttcosttt.xdx2x'2t2t3xt11y由,,k(|1k(x|22xx21xx12x1ykk|)||3211kk2112()2fxetankkxxxk11()2e得ekek由f42fx)ff(1x)f(2xx0）由limlim2xx0f(1x)f(x11lim()f(()2f(422x0y24(x即y4x6f2(xx)fx2(x)[f(x2f(x)f(x)2x0(){[()]2()()2()fxfxfxfx2f3x()[2()]23()()23()fxf3xf2xfxf4xf(x)!f(x)則f(x)(n!f(x)f(x)(nf(x)設(shè)(n)n1(nn2nf(x)!f(x)(n)n1f(x2x)f(x)f(x2x)f(x)limlim22f(x)0000x2x0x0x0f(x)(x2x2f(x)4x又2,00f(x)f(x)()()在xfxfx在x0000.f(x)(xx2)(x99)x(x2)(x99)x(xx(x99)x(xx2)(xf(0)(02)(099)(f(x)f(0)(0)limxxxlim((2)(99),fx0x0x0([f(x)]f(x)(x2f(x)2222dy2xf(x)dx2f(x)f(0)0,f'(0)limxx0f(x)f(0)x(,)0,當(dāng)時xx(,0)(f(x)f(0)0(0)C當(dāng)x,(x)x0在x0f1limf(x)limxsinlimf(x)lim(axb)bb0.2xx0x0x0x01x2sinf(x)f(0)f(x)f(0)axx(0)limlimf(0)lima,又f0xx0xxx0x0x0a0111112111e2)e22()2ed）2xxxxxxx2x2xtdytdy2222tt|9t1）3,3,1122tt12x1yyyy11y221(22(yy3yy3y2y3y21ysinxcosxsin2x）2ycos2xsin(2x)y2cos(2x)2sin(2x2)2222sin(2xn)設(shè)y()n1n2則y(n2cos(2xn)2sin(2x(n)nn22y2sin(2x50)2sin2x(50)49492yxx1xyxx)1y1xxxy()[lnxx)1x]1x1xf(x)f(0)(0)limlim(xx2)(x(xfxx0x0f(x)(x)(xa(x)f(a)(a)）()()fxfa()(()()xaxxa()lim又falimxaxaxaxa()()xa()]()()()xaaaxaxa(x)xa在d1fx1)lim[fx1)(x1)]）2x1x1x1sinx12x11limf(cosx1)limf()12x1x10f(x)f(x)x0在x,(0)(0)(),且fx在x0fxfxflim()lim()(0)則ff即x0x0f(x)ba2ab20而x0f(x)0x0f(x)f(0)sinx)a2ba2(0)limlimb又fx0xx0x0e1ba2e1x(0)faxxx0x0x0ab1a由ab20b1(x,y)xyaxyb則20200000xyya2x2yy0y對x22xx2a在xyky(,)|y20001xx0y0bbbyyxyxx2bb(,)|在xyky200xx0x20xbb又kk()1012yx20xy000.xtan設(shè)t,則x500d11407dxsect2dt500dt500257dcos2dt25x500當(dāng)m,4717d當(dāng)x500m,dt25250f(xh)f(x)f(x)f(h)f(x0)()limlimfxhhh0h0f(x)f(h)f(x)f(0)f(h)f(0)limlimf(x)hhh0h0f(x)f(0)f(x)36yx2xk3x26x|3,1x13x,即3xy60y11k32k113(x3yx80即y3limxlnxx02xxfx.fx48x3x34x6x3xy42.fxxx0在2m1xey3x7.若fx的a且則是x,babfxfxa,b在00.x2x1在.y311limcotx()________.sinxxx011lim()_________.x2xtanxx0e2yyxex1xf(x)x(x)f(0)f(0)f(0)lim（）f則x2x001()((,(()fx（設(shè)fxxxx）2;;.(x)(a,b)在x(a,b),f(x)f(x)0f(x)xx則在）f,0000;;(x,f(x)).00(x),(a,b)(a,b)f(x)0內(nèi)(a,b)f(x)f(a)上設(shè)f在ab在）;;.a,b(x)g(x)f(x)g(x)0f(x)g(x)f(x)g(x)axb,則當(dāng)時,則有設(shè)f（.在持續(xù)可導(dǎo),,且）(x)g(x)f(a)g(a)f(x)g(x)fb)gb)f;f(x)f(a)g(x)g(a);.g(x)g(a)f(x)f(a)3310(,)xx;;;.f(x)(x)x0在f(0)0lim2x0f(x)處（）f且,1cosxx0'(0)0;且f;;.f"(x)|x|(x)f'(0)01,設(shè)f且,）x0(0)f(x)是f(0)f(x)是;f;(0,f(0))yf(x);f(0)f(x).,bf(x)0f(x)[a,b]在(a,b)在f'(x)(a,b)則在）設(shè)a,;;;2.在區(qū)間[）1(x)f(x)|x|;）f;）x2）f(x)1x2;D）2f(x)x2x1.(x)(a,b)(a,b)f'(x)0內(nèi)f(x)(a,b)在f）;;;.f(x)yye0f'(x)0f(x)則）設(shè)y且x0）x;）x;00x;x.00arccosxlncotxlimxlimx0;1lnxxee11x)xx;;x2x)xxx02x0xarctanxtan(ax)lim;.x3bx)x0x0aeab.設(shè)bba當(dāng)0xtanx2sinx3x.2xsinxy(0)(6).y30,n與x3x3.)axa與nx(x),(ab)使設(shè)f在ab1b3a33))ff2.baf(a)fb)其體積V.r(x)[在f(0)f0F(x)xf(x)(,,7.若f且,設(shè)3"'()0使F.1lnx1x0limxlnxlimlimlim(x)01x0x0x0x0xx2(,)f(x)2x0f(x)(,)在f(x)24x12x12x(x2)232()02令fxx1x22f(x)0x2f(x)0當(dāng)xf(2)(y4x12x3y12x1212(xx,當(dāng),3,21y0x(y0x)y0當(dāng)x,當(dāng),當(dāng),(1111xx(x224mm(2m22(,e)ye3xee3x),23x3x3x33233)36)9()ye3xe3xe3xxe3xxx322230,y0當(dāng)x時y0令yx當(dāng)x332222ye(,e)x,223333()()()fx()fxfxfx()0"()00fx,fx0xxxxxx00x0x00000,f(x)f(x)xx()(),fxfx當(dāng)x當(dāng)00y3x20y(,)2,在limylimy(,).在又1.xx1cosx(xsinx)xsinxlim1cosx1limcosxlimlim.6xsinxx3x6232x0x0x0x013xxxxx11x122=.x2xx33x23x0x23x0x0x022,)'2222(,"[2(2)]y"0x,當(dāng)x(,)時,"0,yx2e令x2x22222222,)"0(.22(0,)yex1(,)y'e1,.,且,xx(0,)y'0內(nèi)yex1(0,)在.xf(x)x()1fx()fxlimlimlim1x22x2x0x0x0111(f(x)0f(x)4x14(x)0x(當(dāng)x,又2421f(x)(在.2(x)xf'(0)f"(0)00f(x)xf(x)x的拐點;設(shè)選（Ｄ）f,則,x是,則334f'(0)f"(0)0而x0是f(x)x4.(x)(a,b)f(x)0(a,b)f(x)CC內(nèi)（,4.f在內(nèi)f(x)在[a,b]Cf(a)(a,b)上f(x)f(a).()()()()()()()()0fxgxfxgxfxgxfxgx[f(x)0g(x)f(x)g(x)x(a,b),f(x)f(a).g(x)fb)(x)x3x1f(x)3x3xx令f3則2;1f(x)0f(x),當(dāng)x當(dāng)x當(dāng)x而f,,(f(x)0f(x),,)f(x)0f(x),.,(3f1,limf(x)limf(x),xxf(x)(,在[,在)在.(x)ff(x)f(x)lim200x0;1cosx1cosxx0由1cosx0有f(x)0f(0)f(x)x0即在.f"(x)|x|f"(x)|x|1000"()0;fx0xxx0'(x)f'(0)0f'(x)x0x0f(x),是,f.,在(a,b)f'()0使.(x)x0在f(x)x0在ff(.fx(,)f'(0)0.如y由f在但3'(x)0y"(x)ey'(x)e0x0()x有fx在.x00000arccosxx1limx112x1111x21x1xxx2x11(cscx)2lncotxxsinxcotxlimlimlim1.1lnxcosxsinx2x0x0x0xeexe(exxx1x1xxxx2x)x3x33x26x0x0x0x01111x)]limxx)1121x2xlim]xx2x2x)x0x0x0x011xarctanxx11x22limlimlim.x33x23x2x)32x0x0x01(ax)a2tan(ax)bx))bx)(ax)a21tan(ax)bx)b2bx)bx0x0x02bx)()a21()bx02bblnaalnbaba(x)xlnaaxf(x)[a,b]在令f則af(x)lna0x[a,b]xf(x)[a,b]fb)f(a)在得blnaalnbalnaalna0abab,即(x)x2sinx3xx)在令f時211()23fx33xx302xxxx3xx22f(x)0f(x)(0,)limf(x)lim(tanx2sinx3x)0,在又2x0x0x(0,),f(x)limf(x)0tanx2sinx3x,即2x0(0)(0)n()ff(x)f(0)f(0)xxxo(x)f2nn!x3x5x2m1而xx(o(x)m12m(2mx6x8yx3xx4(6)(0)f1x6f(0)(6)nx)x3n1xx1n633x21x333x0x03x2x0an31n61a,2bf(b)af(a)33)ff令F2ba(x)x3f(x)F(x)[a,b]則在Fb)F(a)ba(a,b)F()使bf(b)af(a)33ff))即即23ba1b3a3))]ff2baf(a)fb)hRhrR2rhr2h2r22R2Rh211hr(2)VRh2hr33h2r1(24hrhrh)2dV12hr(h2r)h(h2r)r2223dh3(h2r)22dV令0h4rdh116rr8224rr當(dāng)h此時V334r2r3(xF'(xF"(xF"'(x)[在F(0)F,由羅爾定理,(7.F上消失,又使1F'()0,又F'(0)[3xf(x)xf'(x)]|0,可知F'(x)在]上知足羅爾定理,于是23101x(0,)"()0"(0)[6()6'()"()]|0''()],對Fx在,使F,又Fxfxxfxxfx23上212x02(0,)(0,)("'()0.F21xxdxx2x(x3x2)dx2cos2xdxcosxsinxdx1cos2xsintdttsinxdxxarctanxdxsin2xdx1sin2x()xdx1(xx1dx__________.225xxfxdx.fxB）fxdxfx;）dfxdxdfxfxdxfx.（）dfx）dxdfxdx,fx在則）fxC）fx）fx）fx）.若Fx和Gxfx）FxGx0;Gx0）Fx）FxGxCGxC）Fx）()c則fx（）()若fx3dxx36955cxc3xc）xc）.）x）3355(x)xlnxxf(x)dx（）設(shè)f則1111(lnx)cx(lnx)c）x22B）2;.24421111）x(lnx)cD）x(lnx)c24224(x)dxxcxfx)dx（）設(shè)f2則2x)cx)c;））22）2211x)cx)c.22）2222e1x（）.e1xln|e1|cln|e1|c;）;B）xx2ln|e1|c2ln|e1|xc.）x;）xx(x)sinxf(x)若f則））1sinx;）1sinx;）1cosx;）1cosxf(0)1F(x)xf(x)x..F'(x)f(xf(x)且又f(x)2則=（））2x1;）x12;）2x1;）x12.3223xx（）2x3333x2ln()C3x2x()C）;）1;xx2222323）3()C;D）3x()C.xxln3ln22ln3ln223edx=（）xx11113eC3eC3ex3e.）））xxxxxxxln31ln3ln31ln311secdx=（）2x2x1111tanCtanCcotCcotC;.;;xxxxxx1dx;xx24x13a22x1x2xexx;dx;ex11exsinxdxx2;.01fx.exsincos2tan設(shè)f2xx2x當(dāng)x設(shè)Fx為fx當(dāng)x求fx.0fxFxsin2xF0Fx0且212xAxB212x12x且fx求設(shè)fxfxfx.22535CxxdxxxCx.22255223x3Cdx53xdx2xC.223x2x1313x2xC(x3x2)dxxx2xCx32232.32322xxx22sinxcosxCxxxx(cosxsinx)dxsinxcosxC.12dxdx11tanxCsecxdxtanxC2.1cos2x12cosx1222sint2costC2sintC.2cosdttdttxcosxsinxCxsinxdxxdcosxxcosxcosxdxxcosxsinxC.arctanxarctanxCarctanxdxxarctanxdarctanxxxarctanxarctanxC.sin2x2sinxcosxsin2x)Cdxdx1sinx1sinx22d2xx)C2.1x2xfxCxdxxxfxdx()()()()()()(x)df(x)(x)f(x)Cx12)C121t則xt令x212dt2t2)tt2221ttx12d()2arctan()C2)C1222()12212x11x1arctanC422C.22x5(xx22fxdxfxdxfxCdfxfxC.由dfxdx.,,A）d[f(x)dx]f(x).(x)f.()）fx3dxx3C23()3,()3tfx3x2ft925f(x)f(x)3xxC335xdF(x)xd(xlnx)xlnxxlnxdx）2x2x11xxx(x)C22222411x)dxfx)dx)x)C）xf222222e1e122xx1）e1xe1e1xxx21e(exx2ex(eexxxx11x2()dex2x2ln|e1|Cxxee1xxx2ln|e1|Cx'(x)x()xC,f,fx1f(x)2f(x)dxsinxCxC,1C1取C12F(x)xf(x)x2F'(x)f(x)'(x)2x又F'(x)f(x)f'(x)2,(x)2dx2xCf(0)1Cf(x)2x1.f3223313xx2()3x2()Cxx32x22ln2133x2()C.xln3ln22113edxe)dxe)3e）.xxxxxxln3e1ln31111111secdx()secdxsecdtanC222.x2xx2xxxxx112(ax)d(ax)axCdx222222;2a2x2x112x421d(x4xd(x2)24xx22x24x2x24x(x2)23211x2ln(x4xarctanC;2233x1x2x(1x)211xx1x()221x21x2xxC;xexdx令ex1txln(t則2e1xttt22得tt21t22ttt222t212tln(t4(tarctant)C22e1x4e14e1C;xxx1cos2x121sinxdxxdxxdxxcos2xdxx22211111xxdsin2xxxsin2xcos2xC22;44448e)exxe)d(e)ee)exxxxxex1ex1eexxee)xx1exee)xe)C.xxxx2)212f2xx2x2x1x2x1f(x)12x2x0xsin11xx11f(x)f(x)dx(2x)dxxln|x1|C2x1xx)C2(x)F(x)sin2xf21cos4xf(x)F(x)dxsin2F(x)dF(x)dx2212x1(x)sin4xC28F21(0)1C1F(x)0()得Fxxx41由F知又41112f(x)F(x)(x4x4x)242x)12xAxB12x21B2Bx2x)2Ax12xAxC1B2Bx2x)2由不定積分的界說：有(12xAx2x)2AxAx2A1B2Bx2即2x)22x)22x)2A2B21AB,2A1B33()(0)fxax2bxca(0)00,c(2)04202aba由f由fbf(x)ax2ax2()0令fx02,xx12()22又fxaf(0)2a0x0f(0)2,f(2)2a0x2f(2)6,a(x)f(x)dx(ax2axdxxaxc而f2323a84acba3由3c2c2f(x)x3x232sinx)dx244411xdxsin_________.340x1.1x20x1________.1dxx2021x2.0則設(shè)fx在2dsinxftdt3x.dxf2則.設(shè)fx在22ftdtx3且x1e3.x1x1.xx1212sin31xx4x2cosxdx.1x2'(x)(2)bfxdxfa1sinxdx0111（）n1n2nnn）0）e））1.dsintxdtx若fx則fx.dx0sinx1cosx）sinx）））0.2xex.x26）0）2））.e22tftdt則（）12設(shè)funxdx00））1）2）4.t2fx2xf2fxfx,則.02e2e2xe2）2）ex）2）.xx2sinx1x4cosxdxN(sinxcosx)dx設(shè)M2,2,2222P(xsinxcosx)dx）45222PMMpNNMP0()(),fx是gx的PMN）N;）））.(x)xcos(t)dt,g(x)sinxx2設(shè)f22x0.(x)()ex()F(x)設(shè)f且Fxftdt則）x2ef(e)2xf(x)ef(e)f(x)2;）2;）xxxx(e)2xf(x)ef(e)f(x)2.）ef2;D）xxxx1(x)[a,b]f(x0)()0(a,b)x,ftx9.f,且ft)ba）0;1;2;D.d(x)tf(xt)dt（x設(shè)f則22）dx0(x)xf(x)2xf(x2);）2xf(x)2.）xf2;）2;）1(x)()2且fxxft則()f(x)設(shè)f=（）0）x1;）x1;x1;D）x1.tx2=（）0xx01;;;.244;xx;x3x201xarcsinx12cosx2dx;dxx2211x222sint)xx2limlim;.00x3x2x0x0fx在x12且limfxlimfx997求x12x12x12tfududtlim12t.12x3x12lnt1xdtx0fxf求設(shè)fx.x1t1xlnx1cos2xdx.e0afxf002afx在且20Mmaxfx.axb0gu并xfthx,x,xtftxx6.fx在gx,hx000求hx.321cos23212x)dxsinx)dxdxcos2xdx244442444431432sin2|x.424212334141333(52)1xdxx)dx)x)|2).222412221331125221522xdxx|x|323.44404033300arcsinx112211arcsin(arcsin)(arcsin)|()dxxdxx2102.822281x20012x11111011ln21dxdxln|x1|ln222.x22x21220023123221xdx2x2x1dxxxx22323000114得2xf(x22)1令x2f(2)4dx1xde3e3ln)ln)21lnx2xe312x1lnx11111dxx(xx1ln2(xx)dxlnxln(x2212221111lim(lnxlnx1)0ln2ln2222x2x(x3x[x22x04201x201(x)C[f(2b)f(2a)]f,2'(x)()fxCf1xfudu11b2b(2)令2fxdxu()()|[(2)(2)]fu2bfbfa2222aa2axx|xx|4(2)22222002xxxx(cossin)dx(sincos)dx222202xxxxcos)|(cossin)|]20222224(211lim()n1nnn1111dxlim()1x)ln21012nnn1x0n111nn0dsin(tx)dtd(x)tx)sinx）fxdxdx020202|)xxe||2|2|22ee20202xxxx22012nt21nx令xt得n(2)22()ft()tftdtdt24000n4()2()fxfxN020P02022）M2,200x8x2x2t)dt2f(x)0022cosx142limlimlim）0g(x)x10x5x101088x0x0x0'(x)f(e)(e)'f(x)(x)'ef(e)2xf(x)2.）F22xxxx1(x)ft)xx）Fft)ba1F(a)0(),Fb()0ftabft)ba1()()又Fxfx0()()0Fx使F.f(x)x121212x0fu)22(xt)f(xt)d(xt)xtfu)x2x22222200x201dx2,=fu)duxf(x)2.2dx0f(x)x21ft.,a1t)a,11.Af0012a2aa11xdx,f(x)x1.00costx2x2limlim1）0x1x0x0242sint得令xx3x208sint2cost2costdt32(cosxcostdcost32222001132(xt)532530251413x2524||||xxxxx41511xx1211121x1xxx222211111x22222316(xcosx)dx(x2xcosxcosx)dx2(xcosx)dx2222424222022212x13x2()22cos2x2x)32202232200111133x|2x|x|4x|2020202022488x2132x.0x33x2x0x0t)xx)2x11.0x2x)2x0x0x0[u)duxfu)xfu)xxtxx)32x)22x)2xxxu)(x)xffxfx(x)()()x)19946xx1299761ln1f()xlnt111u1tx()u2dux1tu111ulnulnulntxxxu2uuulnttt1111lntlntlnt[f(x)f()xxxx1ttt1ttt111lnt11xdtlnt|lnx2x12t22112xdx2xdx22x|2220000x11ex(x)lnx22f(x)則xe令feex()0令fxxee)f(x)0f(x)(,)f(x)0f(x)在又,,在內(nèi),xxlimf(x)lim(lnx)limf(x)lim(lnx)eex0x0xx(e)220而ff(x)e)在(,)在xlnx1cos2xdx(0,)在即e0|f(x)|[f(0)f()x|)aax00ax2a2222|f()||f)|||f)a020h(x)xft)t)xx00(x)ft)(x)(x)g(x)x0(x)g(x)xx00(x)|gu)h(x)h(0)gu)xx即hx000(0)0()hxgu()(x)g(x)f(x)x而h0ye,yex及y3xy2xyyx1x2,yxx1.1.2e,yeyxxxf(xxaxbx持續(xù)曲線y直線,及軸所圍圖形繞x軸扭轉(zhuǎn)一周而成的立體的體積v繞yv4axxx(x0)y2xx00a(costtsint)ya(sinttcost),從0txx2xAyx32與x.xyx,yxS0l.era自到cos)0,2y及x,ya,ylnb(0ab)yA（.lnbelnbeln;）edx;）e;）xdx.bb）xylnaealnaea2acosA（.r121(2acos)d(2acos)）2;）2;2201122(2acos)d(2)）2;）a2d.22200,aeAr及（.1aa222）aed;）2ed;）aed;）2ed.222001x)0x2上s（y.21x221111;）））1x2;221x0202x1211)]dx.x;）2221x200(xy)xy）222222cos24cos2））;）;440012cos2d)d.24;）4200x,xyy22繞y）33;;2;.510423xyx從a到b）232322244ba)ba);33;3332323333b)a)]b)a)].22;229x2xy2y2;3;22）1;4.x4x3(0,和.y2a2r22.cosxsinx,y0(0x)y繞.x4x繞.attyat)y0xacos)a0.r1x,xy2y及.xReyeye),xy與及取xS(ee)|1e|1x10x100323y3x2與y2x(取x132S1x)2xdx[3xxx]23213.333121111x1x)dxdxx1xdx21xdx)2S222211111232x)|2211.223e21()[]2e1Seeee10ee1.xxxx0b[f(x2(xb,.2aa022ax0y2Vax2.02000a22dxdtdyatcost,atsint,dta(cos)(sin)Sattattdtatdt2.2220037yx(xx2)xx0,x2,則1212337A(x0x2x)dx(x2x2x)dx3232.121042A2|xx|0(cosxsinx)dx(sinxcosx)dx2(cosxsinx)dx42440441a2(eaa()()a2dSrr)')de22a2001a21aed(e2aaa0由cos)xcos),ycos)sin0得,8時)2)2Vy20d20264cos)sin2cos)d.2220aba)bx)bxS,alnbe.yySlna12[)]d知A21212(2acos)2dad.(2cos)A222202111(ae)ae,Aa2ed..dA222222x1x211212S1x1.221x2.221x20005.）x軸.ycos.2]41dS4S422d.244210011()(11)31Vydyy22dyyy25.251000011'x,1()1.ydsx2dxxdx2222333sb1xdx[x)]b)a)].2b2233aasinxL2xy2,設(shè)Ly221221221'2L1yxdx00cosx(0)y2sin()(2)212xdx02則L22L=L.212034x3y2x6和(y,29.33S(4xdx3(2x6)dx(x4xdx2243002122rdada2.S2222002)xx2)xx解：V解：V2.4442002)[(2)atdattat3223002a3cost3tt)2a3.3230dsr)2rda'()cos)sin2|cos|dad,2222r)acos)2]rr()),r,.,s2ds)4acos8da.2001xx1y在x1yx,xx2x1112A2(x2)dx(xlnx2x)|ln2221.x21[R,R]xx解：取固定直徑為x軸,x為積分變量且xA(x)3(R2x2),過點且垂直于軸的立體截面面積為43RRR(x)3(Rx)23(Rx)于是V2222R3.3RR02xy1xyxxy.2234CeCey2.xx12設(shè)曲線上隨意率性一點(x,y).t(x)f(x)2xf(dtln2f(x)f則2020yyyy.r2r.12）..cec,cy（4y.xc2112,y()()yc(yy)yyPxyQx則12122.若P(x,y)Q(x,y)0P,Q.yyx,yx2.2xyyy3y.y1x2()0yp則y.x0x0yyy2yy.duu22x2y2a2y(edx)00yxy2.arctanx2x2y2(y)xyxyx0(y)3xyx3;322;22330yysinyy2x.yy;d2yy0c,c,c2212xycsinx;yy;cxcxycxcsinx.;122x(x,y)）y;;;.dydy(xy)(xy)cosyxydx(y2xyy)dy0,（3）中,（1）,（2）22dxdx（）;;;.y(x)(0,y2y4xx1y,（）y()sinypxyxxyxx）xycxx;.yxx;yxy;f(x)y2y4y0若f(x)0且f(x)0則f(x)在x）設(shè)y000;;;.()()0若和y是二階齊次線性方程yPxyQxy的兩個特解,c.c為隨意率性常數(shù),則y1212ycycy（）1122.y(x)2xy60yy(x)平行,而知足方程曲線y經(jīng)由原點,且在原點處切線與直線250yyy）ecos2x1yesin2xyecos2x1yesin2x.yxxxxyxy2y）bax2;;axax;2.42yyxy）cos2xaxcos2xx(a2xb2x)acos2xbsin2x;.ax2;;xyycyf(x)(x2y)y2xy.2(xyx)dx(yxy)dy0）22;dydx）xy(lnylnx);）xyyxe;xlnxdy(ylnx)dx0y1;）x,xe1）yyxy6;2x(xy)dx(xy)dy0）2;1）y;1x2）yyx;()y2y;）yy）y2yyxe.x(x)xfu)()()設(shè)fx,fx求fx.0yf()1B[sinxf(x)]dxf(x)dyf(x).xAyy(x),y(x),y(x)()()()yPxyQxyfx,且設(shè)y,12yy21233yc)y(cc)ycyc,c.112122312km1.k2.CeCe對y2xx12yCe2Ce2,yCe4,320Ce2三個式子連立消去CC,yyy.xxxx121212CeCe另解,y2,xx12r2r2r20,從而對應(yīng)的微分方程是分方程的特點根為r,其對應(yīng)的特點方程為12320.yyyyyy(x)y1.xt(x)2xf(dtln2f(x)2f(x)f(x)2xc,解此微分方程得,即對f雙方求導(dǎo)得20tln22求得c,從而f(x)ce2,又由f(x)2xf(dtln2可知,f(0)2,代入f(x)cexx20f(x)2e2.xr20rr2r212cece為y2.xx12r2r24r40r,12440yyy.(y)y0y0.2210（2,yy20,y,,yxc1y0y.xccec,cyceceece,22（3）錯誤,y不是互相自力的,事實上,xc2xccxx11211.,yyy（4）準(zhǔn)確,,y不相等,1212(yy)yc(yy)y.c12122Q(x,y)P(x,y).xyx1yx1和,9.,y2c(xc(xy