高等數(shù)學(xué)積分公式和微積分公式大全

1、常 用 積 分 公 式（一）含有axb的積分 (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的積分10daxb x32()3axbca11dx axb x322(32 )

2、 ()15axbaxbca122dxaxb x222332(15128) ()105a xabxbaxbca13dxxaxb22(2 )3axbaxbca142dxxaxb22232(348)15a xabxbaxbca15dxxaxb1ln(0)2arctan(0)axbbcbbaxbbaxbcbbb162dxxaxbd2axbaxbxbx axb17daxbxxd2xaxbbx axb182daxbxxd2axbaxxx axb（三）含有22xa的積分1922dxxa=1arctanxcaa2022d()nxxa=2221222123d2(1)()2(1)()nnxnxnaxanaxa21

3、22dxxa=1ln2xacaxa（四）含有2(0)axb a的積分222dxaxb1arctan(0)1ln(0)2axcbbabaxbcbabaxb232dxxaxb21ln2axbca2422dxxaxb2dxbxaaaxb252d()xx axb221ln2xcbaxb2622d()xxaxb21daxbxbaxb2732d()xxaxb22221ln22axbacbxbx2822d()xaxb221d2 ()2xxb axbbaxb（五）含有2axbxc (0)a的積分292dxaxbxc222222222arctan(4)44124ln(4)424axbcbacacbacbaxbb

4、accbacbacaxbbac302dxxaxbxc221dln22bxaxbxcaaaxbxc（六）含有22xa(0)a的積分3122dxxa1arshxca22ln()xxac32223d()xxa222xcaxa3322dxxxa22xac34223d()xxxa221cxa35222dxxxa22222ln()22xaxaxxac36222 3d()xxxa2222ln()xxxacxa3722dxxxa221lnxaacax38222dxxxa222xaca x3922dxax22222ln()22xaxaxxac40223() dxax22224223(25)ln()88xxaxa

5、axxac4122dxxax2231()3xac42222dxxax4222222(2)ln()88xaxaxaxxac4322dxaxx2222lnxaaxaacx44222dxaxx2222ln()xaxxacx（七）含有22xa(0)a的積分4522dxxa1archxxcxa=22ln xxac4622 3d()xxa222xcaxa4722dxxxa22xac4822 3d()xxxa221cxa49222dxxxa22222ln22xaxaxxac502223d()xxxa2222lnxxxacxa5122dxxxa1arccosacax52222dxxxa222xaca x532

6、2dxax22222ln22xaxaxxac54223() dxax22224223(25)ln88xxaxaaxxac5522dxxax2231()3xac56222dxxax4222222(2)ln88xaxaxaxxac5722dxaxx22arccosaxaacx58222dxaxx2222lnxaxxacx（八）含有22ax(0)a的積分5922dxaxarcsinxca6022 3d()xax222xcaax6122dxxax22axc62223d()xxax221cax63222dxxax222arcsin22xaxaxca642223d()xxax22arcsinxxcaax6

7、522dxxax221lnaaxcax66222dxxax222axca x6722daxx222arcsin22xaxaxca68223() daxx222243(52)arcsin88xxaxaxaca6922dxaxx2231()3axc70222dxaxx42222(2)arcsin88xaxxaaxca7122daxxx2222lnaaxaxacx72222daxxx22arcsinaxxcxa（九）含有2axbxc(0)a的積分732dxaxbxc21ln 22axbaaxbxcca742daxbxc x224axbaxbxca2234ln 228acbaxbaaxbxcca752

8、dxxaxbxc21axbxca23ln 222baxbaaxbxcca762dxcbxax212arcsin4axbcabac772dcbxaxx2232242arcsin484axbbacaxbcbxaxcaabac782dxxcbxax23212arcsin24baxbcbxaxcaabac（十）含有xaxb或()()xabx的積分79dxaxxb()()ln()xaxbbaxaxbcxb80dxaxbx()()arcsinxaxaxbbacbxbx81d()()xxa bx2arcsinxacbx()ab82()()dxabxx22()()()arcsin44xabbaxaxabxcb

9、x()ab（十一）含有三角函數(shù)的積分83sin dx xcosxc84cos dx xsin xc85tan dx xln cosxc86cot dx xln sinxc87sec dx xln tan()42xcln sectanxxc88csc dx xln tan2xcln csccotxxc892secdx xtan xc902cscdx xcot xc91sectan dxx xsecxc92csccot dxx xcsc xc932sindx x1sin224xxc942cosdx x1sin 224xxc95sindnx x1211sincossindnnnxxx xnn96co

10、sdnx x1211cossincosdnnnxxx xnn97dsinnxx121cos2d1 sin1sinnnxnxnxnx98dcosnxx121sin2d1 cos1cosnnxnxnxnx99cossindmnxx x11211cossincossindmnmnmxxxx xmnmn11211cossincossindmnmnnxxxx xmnmn100sincosdaxbx x11cos()cos()2()2()ab xab xcabab101sinsindaxbx x11sin()sin()2()2()ab xab xcabab102coscosdaxbx x11sin()si

11、n()2()2()ab xab xcabab103dsinxabx2222tan22arctanxabcabab22()ab104dsinxabx222222tan12lntan2xabbacxbaabba22()ab105dcosxabx2arctan(tan)2ababxcababab22()ab106dcosxabxtan12lntan2xababbacabbaxabba22()ab1072222dcossinxaxbx1arctan(tan )bxcaba1082222dcossinxaxbx1tanln2tanbxacabbxa109sindxax x211sincosaxxaxca

12、a1102sindxax x223122cossincosxaxxaxaxcaaa111cosdxax x211cossinaxxaxcaa1122cosdxax x223122sincossinxaxxaxaxcaaa（十二）含有反三角函數(shù)的積分（其中0a）113arcsindxxa22arcsinxxaxca114arcsindxxxa2222()arcsin244xaxxaxca1152arcsindxxxa322221arcsin(2)39xxxaaxca116arccosdxxa22arccosxxaxca117arccosdxxxa2222()arccos244xaxxaxca11

13、82arccosdxxxa322221arccos(2)39xxxaaxca119arctandxxa22arctanln()2xaxaxca120arctandxxxa221()arctan22xaaxxca1212arctandxxxa33222arctanln()366xxaaxaxca（十三）含有指數(shù)函數(shù)的積分122dxax1lnxaca123e daxx1eaxca124e daxxx21(1)eaxaxca125e dnaxxx11ee dnaxnaxnxxxaa126dxxax21ln(ln)xxxaacaa127dnxx ax11dlnlnnxnxnx axaxaa128esi

14、ndaxbx x221e (sincos)axabxbbxcab129ecosdaxbx x221e ( sincos)axbbxabxcab130esindaxnbx x12221e sin( sincos)axnbx abxnbbxab n22222(1)esindaxnn nbbx xab n131ecosdaxnbx x12221e cos( cossin)axnbx abxnbbxab n22222(1)ecosdaxnn nbbx xab n（十四）含有對數(shù)函數(shù)的積分132lndx xlnxxxc133dlnxxxln ln xc134lndnxx x111(ln)11nxxcnn

15、135(ln) dnxx1(ln)(ln)dnnxxnxx136(ln) dmnxxx111(ln)(ln)d11mnmnnxxxxxmm（十五）含有雙曲函數(shù)的積分137sh dx xchxc138ch dx xshxc139th dx xln chxc1402shdx x1sh224xxc1412chdx x1sh224xxc（十六）定積分142cosdnx xsindnx x 0 143cossindmxnx x 0 144coscosdmxnx x0,mnmn145sinsindmxnx x0,mnmn1460sinsindmxnx x0coscosdmxnx x0,2mnmn147ni

16、20sindnx x20cosdnx xni21nnin134 225 3nnninn（n為大于 1 的正奇數(shù)），1i1 133 124 2 2nnninn（n為正偶數(shù)），0i2一、 （系數(shù)不為0 的情況）00101101lim0nnnmmxmanmba xa xanmb xb xbnm二、重要公式（1）0sinlim1xxx（2）10lim 1xxxe（3）lim()1nna ao（4）lim1nnn（5）lim arctan2xx（6）limtan2xarcx（7）lim arccot0 xx（8）lim arccotxx（9）lim0 xxe（10）limxxe（11）0lim1xxx三

17、、下列常用等價無窮小關(guān)系（0 x）sin xxtan xxarcsin xxarctan xx211cos2xxln 1xx1xex1lnxaxa11xx四、導(dǎo)數(shù)的四則運(yùn)算法則uvuvuvu vuv2uu vuvvv五、基本導(dǎo)數(shù)公式0c1xxsincosxxcossinxx2tansecxx2cotcscxxsecsectanxxxcsccsccotxxxxxeelnxxaaa1ln xx1loglnxaxa21arcsin1xx21arccos1xx21arctan1xx21arccot1xx1x12xx六、高階導(dǎo)數(shù)的運(yùn)算法則（1）nnnu xv xu xv x（2）nncu xcux（3）

18、nnnu axba uaxb（4）( )0nnn kkknku xv xc ux vx七、基本初等函數(shù)的n 階導(dǎo)數(shù)公式（1）!nnxn（2）nax bnaxbeae(3)lnnxxnaaa(4)sinsin2nnaxbaaxbn(5) coscos2nnaxbaaxbn(6)11!1nnnnanaxbaxb(7) 11 !ln1nnnnanaxbaxb八、微分公式與微分運(yùn)算法則0d c1d xxdxsincosdxxdxcossindxxdx2tansecdxxdx2cotcscdxxdxsecsectandxxxdxcsccsccotdxxxdxxxd ee dxlnxxd aaadx1ln

19、dxdxx1loglnxaddxxa21arcsin1dxdxx21arccos1dxdxx21arctan1dxdxx21arccot1dxdxx九、微分運(yùn)算法則d uvdudvd cucdud uvvduudv2uvduudvdvv十、基本積分公式kdxkxc11xx dxclndxxcxlnxxaa dxcaxxe dxeccossinxdxxcsincosxdxxc221sectancosdxxdxxcx221csccotsinxdxxcx21arctan1dxxcx21arcsin1dxxcx十一、下列常用湊微分公式積分型換元公式1faxb dxfaxb d axbauaxb11fx

20、xdxfxd xux1lnlnlnfxdxfx dxxlnuxxxxxfee dxfe d exue1lnxxxxfaa dxfad aaxuasincossinsinfxxdxfx dxsinuxcossincoscosfxxdxfx dxcosux2tansectantanfxxdxfx dxtanux2cotcsccotcotfxxdxfx dxcotux21arctanarcnarcn1fxdxftax dtaxxarctanux21arcsinarcsinarcsin1fxdxfx dxxarcsinux十二、補(bǔ)充下面幾個積分公式tanln cosxdxxccotln sinxdxx

21、csecln sectanxdxxxccscln csccotxdxxxc2211arctanxdxcaxaa2211ln2xadxcxaaxa221arcsinxdxcaax22221lndxxxacxa十三、分部積分法公式形如naxx e dx，令nux，axdve dx形如sinnxxdx令nux，sindvxdx形如cosnxxdx令nux，cosdvxdx形如arctannxxdx，令arctanux，ndvx dx形如lnnxxdx，令lnux，ndvx dx形如sinaxexdx，cosaxexdx令,sin ,cosaxuexx均可。十四、第二換元積分法中的三角換元公式(1)2

22、2axsinxat(2) 22axtanxat(3)22xasecxat【特殊角的三角函數(shù)值】（1）sin00（2）1sin62（3）3sin32（4）sin12） （5）sin0（1）cos01（2）3cos62（3）1cos32（4）cos02） （5）cos1（1）tan00（2）3tan63（3）tan33（4）tan2不存在（ 5）tan0（1）cot 0不存在（ 2）cot36（3）3cot33（4）cot02（5）cot不存在十五、三角函數(shù)公式1.兩角和公式sin()sincoscossinababab sin()sincoscossinabababcos()coscossinsinababab cos()coscossin