不定积分习题及答案

1第四章不定积分(A层次)1．xxdxcossin；2．dxxx2112；3．21xxdx；4．xdxx7sin5sin；5．dxxxxarctg1；6．21xxdx；7．arctgxdxx2；8．dxxlncos；9．dxxxxx3458；10．dxxx2831；11．xdxx2cos；12．dxex3；13．xxxdxlnlnln；14．21xedx；15．dxexexx21；16．dxx3sin；17．dxxx1arcsin；18．dxxx321ln；19．dxxxxxsin2cos5sin3cos7；20．dxxxx21ln；21．xdxx35cossin；22．dxxxtgxsincosln；23．dxxx2arccos2110；24．arctgxdxx2；25．dxxxx1122；26．dxxax222；27．dxtgxex221；28．321xxxxdx；29．xxdxsincos2；30．dxxxxxex23sincossincos。(B层次)1．设xf的一个原函数为xxsin，求dxxfx2。2．dxxexxx11；3．dxxxx2lnln1；4．dxxx1ln；5．dxxx32ln；6．dxxx2sinsinln；7．dxexexx2；8．dxxx2321ln；9．xdxxarcsin12；10．dxxxx231arccos；211．323xxdx；12．dxxxarctgx11112；13．dxxxtgx32cos；14．dxxx1ln；15．dxxx1arcsin；16．dxxxarctgx221；17．dxxx2111；18．xbxadx222sincos；19．dxxxxcossin1sin；(C层次)1．设xF为xf的一个原函数，10F，0xF，且当0x时，有212xxexFxfx，求xf。2．设0,12ln0,00,2sinxxxxxxf，求xf的一个原函数。3．设xf为x的连续函数，且满足方程：Cxxdttftdttfxx981816120，求xf及常数C。4．设xxxf1lnln，计算dxxf。5．设2ln1222xxxf，且xxfln，求dxx。6．设xxxxf1,10,1ln且00f，求xf。7．已知xtgxxf222cossin，10x，求xf。8．若xxf2cossin，且210f，求方程0xf的根。9．求dxxfxfxfxfxf32。10．dxxx1,,max23。第四章不定积分(A层次)31．xxdxcossin解：原式Ctgxtgxtgxddxtgxxlnsec22．dxxx2112解：原式Cxxxdxxxdarcsin1211122223．21xxdx解：原式Cxxdxxx2ln1ln31211131Cxx12ln314．xdxx7sin5sin解：原式xdxxdxdxxx12cos212cos212cos12cos21Cxx12sin2412sin415．dxxxxarctg1解：原式Cxarctgxarctgdxarctgdxxxarctg222126．21xxdx解：dttttttttttdttxxxdxsincossincossincos21cossincossin12令Ctttttttddtcossinln2121cossincossin2121Cxxx21ln21arcsin217．arctgxdxx2解：原式dxxxarctgxxxarctgxd23331131314231313131xxdxxdxarctgxxCxxarctgxx2231ln6161318．dxxlncos解：原式dxxxxxx1lnsinlncosdxxxxlnsinlncosxxdxxxxlnsinlnsinlncosdxxxxxxxlncoslnsinlncos故Cxxxxdxxlnsinlncos21lncos9．dxxxxx3458解：原式dxxxxxdxxx32281dxxdxxdxxxxx13148213123Cxxxxxx1ln31ln4ln821312310．dxxx2831解：原式ttdttgtuuduuxxxd42224284secsec41141141令令dtttdt2cos181cos412Ctt2sin16181Cuuuarctgu221118181Cxxarctgx844188111．xdxx2cos解：原式dxxx22cos1xxdxxdxxxdx2sin41412cos2125xdxxxx2sin412sin41412Cxxxx2cos812sin4141212．dxex3解：令tx3，则3tx，dttdx23原式tdteetdetdttetttt2333222dteteettdeetttttt636322Ceteetttt6632Cxxex2223332313．xxxdxlnlnln解：原式Cxxxdxxxdlnlnlnlnlnlnlnlnlnlnln14．21xedx解：222111111tdtdtttttttteedxxx令Ctttttddttt111ln111112CeexCeeexxxxx111ln111ln15．dxexexx21解：原式11112xxxexdeexdxxxxxxxxedeeexdxeeeex111111Ceeexxxx1lnln1Ceexexxx1ln1616．dxx3sin解：令tx3，则3tx，dttdx23原式tdtdtttcos33sin22ttdtttdttttsin6cos32cos3cos322tdtttttsin6sin6cos32Ctttttcos6sin6cos32Cxxxxx333332cos6sin6cos317．dxxx1arcsin解：令uxsin，则ux2sin，uduudxcossin2原式uduuuucossin2cosuduuuuducoscos2cos2CxxxCuuu2arcsin12sin2cos218．dxxx321ln解：原式22211lnxdxdxxxxxx2222122121ln2222212121lnxxdxxx222221112121lndxxxxxCxxxx22221lnln2121lnCxxxx2221ln21ln21ln19．dxxxxxsin2cos5sin3cos77解：原式dxxxxxxxsin2cos5sin5cos2sin2cos5dxxxxxsin2cos5sin5cos21Cxxxsin2cos5ln20．dxxxx21ln解：原式xdxx11lndxxxxxx1111lndxxxxx11lnCxxxxln1ln21．xdxx35cossin解：原式xdxxxcoscossin25xdxxsinsin1sin25Cxx86sin81sin6122．dxxxtgxsincosln解：原式tgxdtgxtgxdxxtgxtgxlncosln2Ctgxtgxtgxd2ln21lnln23．dxxx2arccos2110解：原式xdxarccos21021arccos2CCxxararccos2cos21010ln211010ln12124．arctgxdxx2解：原式331xarctgxd8dxxxarctgxx2331131dxxxxxarctgxx23313131231313131xxdxxdxarctgxxCxxarctgxx2231ln61613125．dxxxx1122解：令tx1，dttdx21原式dttttt222111111dttttdtdttt2221111Ctt21arcsinCxxx11arcsin226．dxxax222解：令atgtx，tdtadx2sec原式dttattgata222secsecdtttttttdtcossincossincossin2222dttttdt2sincossecCttgttsin1seclnCxxaaxaxa2222lnCxaxax2222ln927．dxtgxex221解：原式dxtgxxex2sec22tgxdxexdxexx2222sectgxdxedtgxexx222dxtgxedxetgxtgxexxx22222Ctgxex228．321xxxxdx解：原式3312421xdxxdxxdxCxxx1ln3ln32ln421Cxxx34312ln2129．xxdxsincos2解：令txtg2，则arctgtx2，212tdtdx，212sinttx，2211costtx，于是原式dtttt3122dtttt313322dttttd131333122Ctt3ln313Cxtgxtg232ln31330．dxxxxxex23sincossincos。10解：原式xdxtgxexdxxexxseccossinsinxdexdxexxsecsinsinsinxxxexdxxdesinsinsinsecsecxdxxexedxexexxxxcossecsecsinsins