经济数学基础-----微积分----第六章习题解答

习题六答案)0()1(2adxeaax2、利用定积分性质，估计下列积分：,)(],,[2xexfaax解：aaxdxe2,1)0(maxf,)(2minaeafdxx4542)sin1()2(,sin1)(],45,4[2xxfx解：,2)2(maxf,1)(minfdxx4542)sin1(22aaea22dxxx24sin)3(,sin)(],2,4[xxfx解：,1)2(maxf,22)4(minfdxxx24sindxx24122dxx2412lnsin2ln2224dxxx即dxxx333arctan)4(,arctan)(],3,33[xxfx解：,3)3(maxf,6)33(minfdxxx333arctandxx3336dxx333394arctan92333dxxx即,11)(],1,0[xxfx解：,1)0(maxf,22)1(minfdxxx1091dxx10922dxx1091011202109dxxx即dxxx1091)5(,201)(],200,0[xxfx解：,201)0(maxf,2201)200(minfdxexx20005201dxex200052201dxex20005201)1(1001201)1(110011000200051000edxexex即dxexx20005201)6(duuxuxx100)2sin1(1lim)1(3、求下列极限：xxx10)2sin1(lim222sin2sin10)2sin1(limxxxxx2eduuxxx020arctan1lim)2(xxx2arctanlim021xxxxx20sec)sin(tancos)tan(sinlimxx30coslim1xxduux020cos1lim)3(20coslimxx1)sin()tan(lim)4(tan0sin00duuduuxxx.)2)(1()(420的极值点、求函数dueuuxfux2)2)(1()(xexxxf解：,02,121xx])23[()(22xexxxf22)23(2)32(2xxexxxex2)3262(23xexxx0)1(1ef0)2(4ef1x极大值点2x极小值点5、利用牛—莱公式计算下列积分：dxxx1)1()1(241xdx241)1(2)1()1(2241xdx413)1(32x32dxxxx213231)2(dxxxxxx2122)1()1)(1(dxxx212)111(21)ln1(xxx2ln23dxxx65221)3(2021ln21dxxex20)()4(dxxx)211(216565)2ln(ln21xx202)21(xex32e)()5(badxxba,0ab解：dxxba原式bax221)(2122ab,0abdxxdxxba00原式baxx02022121)(2122ba,0abdxxba原式bax221)(2122badxbaxdxxbabbabaa22)](2[]2)[(原式bbabaaxbaxxxba2222])([])[(4)(4)(22baba2)(2ba时解：当ba时当ba2)(2badxxbadxbaxbbabaa22]2)[()](2[原式dxbaxba)(2)6(6、用换元法计算下列积分：dxxx)2cos(cos212020)2sin41sin21(xx21dxex2ln01)2(1,12tetexx解：设12),1ln(22ttdtdxtxdtttt10212原式10)arctan(2tt22)111(21022dtttdxxx23cos2cos)1(20dxxx5121)3(tdtxdxtxtx,1,1222解：设dxxxx51221原式2021tdttt2022111dttt20)arctan(tt2arctan2dxxx1sin1)4(212xdx11sin21211cosx1dxxx941)5(tdtdxtxtx2,,2解：设tdttt2132原式dttt3221112322)1ln22(ttt2ln27dxxx221)1(1)6(tdtdxtxtx2,1,12解：设tdttt2210)2(2原式10246)44(2dtttt10357)345471(2ttt105584105478dxxx112521)7()1(2)1(11122xdx1121arctan21x8dxx205)1()8(dxxdxx215105)1()1()1()1()1()1(215105xdxxdx2161066)1(6)1(xx31dxxx16091)9(160)9(91dxxx16016091)9(991dxxxdx16023160233291)9(3291xx12dxxxe31ln11)10(xdxelnln1131)ln1(ln1131xdxe31ln12ex2dxxx)1(sectan240xdx403tan)11(dxxxdxx40240tansectan4040coslntantanxxxd22lntan21402x22ln21)2ln1(21dxxx4142)12(dxxx414222441dxxdxx4141421214241)24(4216141xdx4141234241)24(32161xx223)24(42214141xdx2.1.4.22.cos,sintxtxtdtdxtx设dxxx122221)13(dttt2422sincosdtt242cotdtt242)1(csc24)cot(tt41xxxdeee5ln01312,5ln,0,0,2,1txtxtdtdeetxx设tdttt24202原式dttt202242dttt20224442dtt202)441(220)2arctan42(tt4dxeeexxx5ln013)14(dxxxee1121)1ln(1)15()1ln(])1ln(1[112xdxee1121122)1(ln21)1ln(eeeexx25231dxxxx10232)13(1)16(dxxxx10232)13(3331)13()13(13131023xxdxx154103)13(131xxdxxx20234)17(22022421dxxx2202244421dxxx2202)441(21dxx2022))4ln(4(21xx)2ln1(21dxxxxx22sin2cos)18(dxxxxx22sin2cos2221)sin2(sin2121222xxdxx22sin2ln21xx2ln)cos(sin)cos(11122xdx11sincosarctansin1x)sincos1arctansincos1(arctansin1)sincos1arctansincos1(arctansin1sin2)2cotarctan2tan(arctansin1dxxx1121cos21)19(删去dxxxxx403)cos3sin2(sin3cos2)20()cos3sin2()cos3sin2(1403xxdxx402)cos3sin2(121xx)912(211817dxxxx2102412arctan8)21(dxxxdxxx21022102412arctan418xdxxdx2arctan2arctan21)14(41121022102210221022arctan41)14ln(xx642ln2dxxxxx412)(21)22(dxxxxx412)(2212)()(12412xxdxx4112xx32dxxxx)121()(12412dxxx1041)23(21022)(1121dxx102arctan21x83,8;2,3,1,122txtxdxxxdtxt则设32221ttdt原式32221t11tdt322321t1dtdt32)11ln21(ttt23ln211dxxxxxdxxxx83222832)11)1(11)24(23ln211)24()25(类似，原式与dxxxxe122ln)26(21222ln21dxxxxe)ln(21212221dxxxdxee)lnln(2121212xdxxee)ln211(211222exe)1(212edxxx40coslntan)27(xdxcoslncosln40402cosln21x2)2(ln81dxxxxxx242)sin(sincos)28()sin()sin(1242xxdxx24sin1xx)122(2222212sin,2sec21,2tanttxdxxdtxt则设dxxx2sec2sin181242arctan22原式dttt42221)1(41213)312(41ttt9655dxxx2arctan222sin)cos1(1)29(删去dxx25411)30(5,25,2,4,2,txtxtdtdxxt设dttt52211原式dttt5212dttt521112dtt52)111(2)2ln3(2)]1ln([252tt1,1,0,0,2,txtxtdtdxxt设dxex10)31(dttet102原式10t1)e-2(t27.用分部积分法计算下列积分：103)1()1(dxxx103)1(3ln1xdx]3)1(3[3ln11010dxxxx)33ln11(3ln110x2)3(ln23ln40sin)2(xdxx40cosxxd4040coscosxdxxx40)cos(sinxxx)41(2210)1ln()3(dxx10101)1ln(dxxxxx10)]1ln([2lnxx12ln2102)1ln()4(dxx102210