考研数学常用积分公式

常用积分公式（一）含有axb的积分(0a)1．dxaxb＝1lnaxbCa2．()daxbx＝11()(1)axbCa（1）3．dxxaxb＝21(ln)axbbaxbCa4．2dxxaxb＝22311()2()ln2axbbaxbbaxbCa5．d()xxaxb＝1lnaxbCbx6．2d()xxaxb＝21lnaaxbCbxbx7．2d()xxaxb＝21(ln)baxbCaaxb8．22d()xxaxb＝231(2ln)baxbbaxbCaaxb9．2d()xxaxb＝211ln()axbCbaxbbx（二）含有axb的积分10．daxbx＝32()3axbCa11．dxaxbx＝322(32)()15axbaxbCa12．2dxaxbx＝222332(15128)()105axabxbaxbCa13．dxxaxb＝22(2)3axbaxbCa14．2dxxaxb＝22232(348)15axabxbaxbCa15．dxxaxb＝1ln(0)2arctan(0)axbbCbbaxbbaxbCbbb16．2dxxaxb＝d2axbaxbxbxaxb17．daxbxx＝d2xaxbbxaxb18．2daxbxx＝d2axbaxxxaxb（三）含有22xa的积分19．22dxxa=1arctanxCaa20．22d()nxxa=2221222123d2(1)()2(1)()nnxnxnaxanaxa21．22dxxa=1ln2xaCaxa（四）含有2(0)axba的积分22．2dxaxb＝1arctan(0)1ln(0)2axCbbabaxbCbabaxb23．2dxxaxb＝21ln2axbCa24．22dxxaxb＝2dxbxaaaxb25．2d()xxaxb＝221ln2xCbaxb26．22d()xxaxb＝21daxbxbaxb27．32d()xxaxb＝22221ln22axbaCbxbx28．22d()xaxb＝221d2()2xxbaxbbaxb（五）含有2axbxc(0)a的积分29．2dxaxbxc＝222222222arctan(4)44124ln(4)424axbCbacacbacbaxbbacCbacbacaxbbac30．2dxxaxbxc＝221dln22bxaxbxcaaaxbxc（六）含有22xa(0)a的积分31．22dxxa＝1arshxCa＝22ln()xxaC32．223d()xxa＝222xCaxa33．22dxxxa＝22xaC34．223d()xxxa＝221Cxa35．222dxxxa＝22222ln()22xaxaxxaC36．2223d()xxxa＝2222ln()xxxaCxa37．22dxxxa＝221lnxaaCax38．222dxxxa＝222xaCax39．22dxax＝22222ln()22xaxaxxaC40．223()dxax＝22224223(25)ln()88xxaxaaxxaC41．22dxxax＝2231()3xaC42．222dxxax＝4222222(2)ln()88xaxaxaxxaC43．22dxaxx＝2222lnxaaxaaCx44．222dxaxx＝2222ln()xaxxaCx（七）含有22xa(0)a的积分45．22dxxa＝1archxxCxa=22lnxxaC46．223d()xxa＝222xCaxa47．22dxxxa＝22xaC48．223d()xxxa＝221Cxa49．222dxxxa＝22222ln22xaxaxxaC50．2223d()xxxa＝2222lnxxxaCxa51．22dxxxa＝1arccosaCax52．222dxxxa＝222xaCax53．22dxax＝22222ln22xaxaxxaC54．223()dxax＝22224223(25)ln88xxaxaaxxaC55．22dxxax＝2231()3xaC56．222dxxax＝4222222(2)ln88xaxaxaxxaC57．22dxaxx＝22arccosaxaaCx58．222dxaxx＝2222lnxaxxaCx（八）含有22ax(0)a的积分59．22dxax＝arcsinxCa60．223d()xax＝222xCaax61．22dxxax＝22axC62．223d()xxax＝221Cax63．222dxxax＝222arcsin22xaxaxCa64．2223d()xxax＝22arcsinxxCaax65．22dxxax＝221lnaaxCax66．222dxxax＝222axCax67．22daxx＝222arcsin22xaxaxCa68．223()daxx＝222243(52)arcsin88xxaxaxaCa69．22dxaxx＝2231()3axC70．222dxaxx＝42222(2)arcsin88xaxxaaxCa71．22daxxx＝2222lnaaxaxaCx72．222daxxx＝22arcsinaxxCxa（九）含有2axbxc(0)a的积分73．2dxaxbxc＝21ln22axbaaxbxcCa74．2daxbxcx＝224axbaxbxca2234ln228acbaxbaaxbxcCa75．2dxxaxbxc＝21axbxca23ln222baxbaaxbxcCa76．2dxcbxax＝212arcsin4axbCabac77．2dcbxaxx＝2232242arcsin484axbbacaxbcbxaxCaabac78．2dxxcbxax＝23212arcsin24baxbcbxaxCaabac（十）含有xaxb或()()xabx的积分79．dxaxxb＝()()ln()xaxbbaxaxbCxb80．dxaxbx＝()()arcsinxaxaxbbaCbxbx81．d()()xxabx＝2arcsinxaCbx()ab82．()()dxabxx＝22()()()arcsin44xabbaxaxabxCbx()ab（十一）含有三角函数的积分83．sindxx＝cosxC84．cosdxx＝sinxC85．tandxx＝lncosxC86．cotdxx＝lnsinxC87．secdxx＝lntan()42xC＝lnsectanxxC88．cscdxx＝lntan2xC＝lncsccotxxC89．2secdxx＝tanxC90．2cscdxx＝cotxC91．sectandxxx＝secxC92．csccotdxxx＝cscxC93．2sindxx＝1sin224xxC94．2cosdxx＝1sin224xxC95．sindnxx＝1211sincossindnnnxxxxnn96．cosdnxx＝1211cossincosdnnnxxxxnn97．dsinnxx＝121cos2d1sin1sinnnxnxnxnx98．dcosnxx＝121sin2d1cos1cosnnxnxnxnx99．cossindmnxxx＝11211cossincossindmnmnmxxxxxmnmn＝11211cossincossindmnmnnxxxxxmnmn100．sincosdaxbxx＝11cos()cos()2()2()abxabxCabab101．sinsindaxbxx＝11sin()sin()2()2()abxabxCabab102．coscosdaxbxx＝11sin()sin()2()2()abxabxCabab103．dsinxabx＝2222tan22arctanxabCabab22()ab104．dsinxabx＝222222tan12lntan2xabbaCxbaabba22()ab105．dcosxabx＝2arctan(tan)2ababxCababab22()ab106．dcosxabx＝tan12lntan2xababbaCabbaxabba22()ab107．2222dcossinxaxbx＝1arctan(tan)bxCaba108．2222dcossinxaxbx＝1tanln2tanbxaCabbxa109．sindxaxx＝211sincosaxxaxCaa110．2sindxaxx＝223122cossincosxaxxaxaxCaaa111．cosdxaxx＝211cossinaxxaxCaa112．2cosdxaxx＝223122sincossinxaxxaxaxCaaa（十二）含有反三角函数的积分（其中0a）113．arcsindxxa＝22arcsinxxaxCa114．arcsindxxxa＝2222()arcsin244xaxxaxCa115．2arcsindxxxa＝322221arcsin(2)39xxxaaxCa116．arccosdxxa＝22arccosxxaxCa117．arccosdxxxa＝2222()arccos244xaxxaxCa118．2arccosdxxxa＝322221arccos(2)39xxxaaxCa119．arctandxxa＝22arctanln()2xaxaxCa120．arctandxxxa＝221()arctan22xaaxxCa121．2arctandxxxa＝33222arctanln()366xxaaxaxCa（十三）含有指数函数的积分122．dxax＝1lnxaCa123．edaxx＝1eaxCa124．edaxxx＝21(1)eaxaxCa125．ednaxxx＝11eednaxnaxnxxxaa126．dxxax＝21ln(ln)xxxaaCaa127．dnxxax＝11dlnlnnxnxnxaxaxaa128．esindaxbxx＝221e(sincos)axabxbbxCab129．ecosdaxbxx＝221e(sincos)axbbxabxCab130．esindaxnbxx＝12221esin(sincos)axnbxabxn