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1、学习必备欢迎下载常 用 积 分 公 式（一）含有axb的积分 (0a)1dxaxb1ln axbCa2() daxbx11()(1)axbCa（1）3dxxaxb21(ln)axbbaxbCa42dxxaxb22311()2 ()ln2axbb axbbaxbCa5d()xx axb1lnaxbCbx62d()xxaxb21lnaaxbCbxbx72d()xxaxb21(ln)baxbCaaxb822d()xxaxb231(2 ln)baxbbaxbCaaxb92d()xx axb211ln()axbCb axbbx（二）含有axb的积分10daxb x32()3axbCa11dx axb x
2、322(32 )()15axbaxbCa122dxaxb x222332(15128)()105a xabxbaxbCa13dxxaxb22(2 )3axbaxbCa精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 1 页，共 12 页学习必备欢迎下载142dxxaxb22232(348)15a xabxbaxbCa15dxxaxb1ln(0)2arctan(0)axbbCbbaxbbaxbCbbb162dxxaxbd2axbaxbxbx axb17daxbxxd2xaxbbx axb182daxbxxd2axbaxxx axb（三）含有22xa的积
3、分1922dxxa=1arctanxCaa2022d()nxxa=2221222123d2(1)()2(1)()nnxnxnaxanaxa2122dxxa=1ln2xaCaxa（四）含有2(0)axb a的积分222dxaxb1arctan(0)1ln(0)2axCbbabaxbCbabaxb232dxxaxb21ln2axbCa精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 2 页，共 12 页学习必备欢迎下载2422dxxaxb2dxbxaaaxb252d()xx axb221ln2xCbaxb2622d()xxaxb21daxbxbaxb27
4、32d()xxaxb22221ln22axbaCbxbx2822d()xaxb221d2 ()2xxb axbbaxb（五）含有2axbxc (0)a的积分292dxaxbxc222222222arctan(4)44124ln(4)424axbCbacacbacbaxbbacCbacbacaxbbac302dxxaxbxc221dln22bxaxbxcaaaxbxc（六）含有22xa(0)a的积分3122dxxa1arshxCa22ln()xxaC32223d()xxa222xCaxa3322dxxxa22xaC34223d()xxxa221Cxa精选学习资料 - - - - - - - -
5、- 名师归纳总结 - - - - - - -第 3 页，共 12 页学习必备欢迎下载35222dxxxa22222ln()22xaxaxxaC362223d()xxxa2222ln()xxxaCxa3722dxxxa221lnxaaCax38222dxxxa222xaCa x3922dxax22222ln()22xaxaxxaC40223() dxax22224223(25)ln()88xxaxaaxxaC4122dxxax22 31()3xaC42222dxxax4222222(2)ln()88xaxaxaxxaC4322dxaxx2222lnxaaxaaCx44222dxaxx2222ln
6、()xaxxaCx（七）含有22xa(0)a的积分4522dxxa1archxxCxa=22ln xxaC46223d()xxa222xCaxa4722dxxxa22xaC精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 4 页，共 12 页学习必备欢迎下载4822 3d()xxxa221Cxa49222dxxxa22222ln22xaxaxxaC50222 3d()xxxa2222lnxxxaCxa5122dxxxa1arccosaCax52222dxxxa222xaCa x5322dxax22222ln22xaxaxxaC54223() dxax
7、22224223(25)ln88xxaxaaxxaC5522dxxax22 31()3xaC56222dxxax4222222(2)ln88xaxaxaxxaC5722dxaxx22arccosaxaaCx58222dxaxx2222lnxaxxaCx（八）含有22ax(0)a的积分5922dxaxarcsinxCa60223d()xax222xCaax精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 5 页，共 12 页学习必备欢迎下载6122dxxax22axC6222 3d()xxax221Cax63222dxxax222arcsin22xax
8、axCa64222 3d()xxax22arcsinxxCaax6522dxxax221lnaaxCax66222dxxax222axCa x6722daxx222arcsin22xaxaxCa68223() daxx222243(52)arcsin88xxaxaxaCa6922dx axx2231()3axC70222dxaxx42222(2)arcsin88xaxxaaxCa7122daxxx2222lnaaxaxaCx72222daxxx22arcsinaxxCxa（九）含有2axbxc(0)a的积分732dxaxbxc21ln 22axbaaxbxcCa精选学习资料 - - - - -
9、 - - - - 名师归纳总结 - - - - - - -第 6 页，共 12 页学习必备欢迎下载742daxbxc x224axbaxbxca2234l n 228acbaxbaa xb xcCa752dxxaxbxc21axbxca23l n 222ba xbaa xb xcCa762dxcbxax212arcsin4axbCabac772dcbxaxx2232242arcsin484axbbacaxbcbxaxCaabac782dxxcbxax23212arcsin24baxbcbxaxCaabac（十）含有xaxb或()()xabx的积分79dxaxxb()()ln()xaxbbaxa
10、xbCxb80dxaxbx()()arcsinxaxaxbbaCbxbx81d()()xxabx2arcsinxaCbx()ab82()()dxabxx22()()()arcsin44xabbaxaxabxCbx()ab（十一）含有三角函数的积分83sin dx xcos xC精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 7 页，共 12 页学习必备欢迎下载84cos dx xsin xC85tan dx xln cosxC86cot dx xln sinxC87sec dx xln tan()42xCln sectanxxC88csc dx x
11、ln tan2xCln csccotxxC892secdx xtanxC902cscdx xcotxC91sec tan dxx xsecxC92csccot dxx xcscxC932sindx x1sin224xxC942cosdx x1sin224xxC95sindnx x1211sincossindnnnxxx xnn96cosdnx x1211cossincosdnnnxxx xnn97dsinnxx121cos2d1 sin1sinnnxnxnxnx98dcosnxx121sin2d1 cos1cosnnxnxnxnx99cossindmnxx x11211cossincossin
12、dmnmnmxxxx xmnmn11211cossincossindmnmnnxxxx xmnmn100sincosdaxbx x11cos()cos()2()2()ab xab xCabab精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 8 页，共 12 页学习必备欢迎下载101sinsindaxbx x11sin()sin()2()2()ab xab xCabab102coscosdaxbx x11sin()sin()2()2()ab xab xCabab103dsinxabx2222tan22arctanxabCabab22()ab104ds
13、inxabx222222tan12lntan2xabbaCxbaabba22()ab105dcosxabx2arctan(tan)2ababxCababab22()ab106dcosxabxtan12lntan2xababbaCabbaxabba22()ab1072222dcossinxaxbx1arctan(tan )bxCaba1082222dcossinxaxbx1tanln2tanbxaCabbxa109sindxax x211sincosaxxaxCaa1102sindxax x223122cossincosxaxxaxaxCaaa111cosdxax x211cossinaxxax
14、Caa1122cosdxax x223122sincossinxaxxaxaxCaaa（十二）含有反三角函数的积分（其中0a）113arcsindxxa22arcsinxxaxCa精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 9 页，共 12 页学习必备欢迎下载114arcsindxxxa2222()arcsin244xaxxaxCa1152arcsin dxxxa322221arcsin(2)39xxxaaxCa116arccosdxxa22arccosxxaxCa117arccos dxxxa2222()arccos244xaxxaxCa11
15、82arccos dxxxa322221arccos(2)39xxxaaxCa119arctandxxa22arctanln()2xaxaxCa120arctandxxxa221()arctan22xaaxxCa1212arctandxxxa33222arctanln()366xxaaxaxCa（十三）含有指数函数的积分122dxax1lnxaCa123e daxx1eaxCa124e daxxx21(1)eaxaxCa125e dnaxxx11ee dnaxnaxnxxxaa126dxxax21ln(ln)xxxaaCaa127dnxx ax11dlnlnnxnxnx axaxaa128es
16、indaxbx x221e (sincos)axabxbbxCab129ecosdaxbx x221e ( sincos)axbbxabxCab精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 10 页，共 12 页学习必备欢迎下载130esindaxnbx x12221e sin(sincos)axnbx abxnbbxab n22222(1)e sindaxnn nbbx xab n131ecosdaxnbx x12221e cos(cossin)axnbx abxnbbxab n22222(1)e cosdaxnn nbbx xab n（十四）
17、含有对数函数的积分132lndx xlnxxxC133dlnxxxln ln xC134lndnxx x111(ln)11nxxCnn135(ln) dnxx1(ln)(ln)dnnxxnxx136(ln) dmnxxx111(ln)(ln)d11mnmnnxxxxxmm（十五）含有双曲函数的积分137sh dx xchxC138ch dx xshxC139th dx xlnchxC1402shdx x1sh224xxC1412chdx x1sh224xxC（十六）定积分142cosdnx xsindnx x 0 143cossindmxnx x 0 144coscosdmxnx x0,mnmn精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 11 页，共 12 页学习必备欢迎下载145sinsindmxnx x0,mnmn1460sinsindmxnx x0coscosdmxnx x0,2mnmn147nI20sindnx x20cosdnx xnI21nnIn1342253nnnInn（n为大于 1 的正奇数），1I1 133 124 2 2nnnInn（n为正偶数），0I2精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 12 页，共 12 页
