高等数学常用积分公式查询表

1、.导数公式：(tgx)sec2 x(arcsin x)11x2(ctgx)csc2 x(arccos x)1(secx)secx tgx1x2(cscx)cscx ctgx( arctgx )1(a x )a x ln a1 x21( arcctgx )1(log a x)1x2x ln a基本积分表：tgxdxctgxdxsecxdxcscxdxdx22axdxx2a2dx22axa2x2ln cosxcln sin xcln secxtgxcln cscxctgxc1 arctg x caa1 ln xac2a xa1 ln a x c 2a a xarcsin xcadxsec2 xdx

2、tgx ccos2 xdxcsc2 xdxctgxcsin 2 xsecx tgxdxsecxccscxctgxdxcscxca xdxa xcln ashxdxchxcchxdxshxcdxln( xx2a 2 )cx 2a22sin n2cosn xdxn 1i nxdxi n 200nx2a2dxxx2a2a222c2ln( xxa )2x2a 2 dxxx2a2a 2 ln xx 2a2c22a2x2dxxa2x2a2xc2arcsina2三角函数的有理式积分：.2u1u2x2dusin x 1u 2 ， cos x1u2 ，utg2，dx1u2（一）含有 axb 的积分 ( a0)1

3、dx 1 ln axbcaxba2(ax b) dx a(1(ax b) 1c （1）1)3xdx 1(axbb ln axb )caxba24x2dx 11 ( axb)22b(axb)b2 ln axbcaxba325dxb)1 ln axbcx(axbx6dx1aax bcx2 (ax b)bxb2 lnx7x2 dx 12(ln axbaxb)c( axb)ab8x21b2( axb)2 dxa3( axb2bln axbaxb)c9dxb)21b)12 ln ax bcx(axb(axbx（二）含有axb 的积分10axbdx 2(axb)3c3a11x axbdx 22 (3ax2

4、b)(axb)3c15a12x2axbdx 2 3 (15a2 x212abx8b2 )(axb)3c105a13xx22 (ax2b)axbcdaxb3a.14x2dx 2(3a2 x24abx 8b2 ) ax b caxb15a31axbbc(b0)dxlnaxbb15bx axb2arctanaxbc(b0)bb16dxaxbadxx2axbbx2bx axb17axbdx 2axbbxdxbxax18axbdx axbaxdxbx2x2ax（三）含有 x2a2的积分19dx1xcx2a2=arctanaadxx2n 3dx20( x2a2 )n = 2( n 1)a 2( x2a2

5、) n 12( n 1) a2( x2a2 )n 121dx1xacx2a2=lnxa2a（四）含有 ax2b( a0) 的积分1arctana xc( b0)22dxabbax2b1axblnc(b0)2abaxb23xdx 1ln ax2bcax2b2a.x2xbdx242dx aax2baxba25dx 1 lnx2bcx( ax2b)2bax226dx1adxbxbax 2bx2 (ax2b)273dxaax2b12c(ax2b)2 lnx22bxx2b28dxx1dx2b(ax 2b)2bax 2b(ax2b)2（五）含有 ax2bxc (a0) 的积分2arctan2axbc( b

6、24ac)29dx4acb24acb2bx12axbb24acax2cln(b24ac)b22axbb2c4ac4ac30xdx 1 ln ax2bx cbax 2dxax2bxc2a2abx c（六）含有x2a2(a0) 的积分31dxa2 arsh xc1 ln( xx2a2 ) cx2a32dxxc( x2a 2 x2a2a2 )333xa2dx x2a2cx234xdx 1c( x2x2a2a 2 )3.35x2xx2a2a2ln( xx2a2)cx2a2dx 2236x2dx xa2ln( xx2a2 )c( x2a2 )3x237xdx 1 lnx2a2acx 2a2ax38dxx

7、2a2cx2x2a2a2 x39x2a2 dx xx2a2a2ln( xx2a2 )c2240( x2a2 )3 dx x (2 x25a2 )x2a23a4 ln( xx2a2 ) c8841x x2a2 dx 1( x2a2 )3c342x2x2a2dxx(2 x2a2)x2a2a4x2a2) c8ln( x843x2a2 dx x2a 2a lnx2a2acxx44x2a2dx x2a2ln( xx2a2 )cx2x（七）含有x2a2(a0) 的积分45dxa2 x arch xc1 = ln xx2a 2cx2xa46dxxc( x2a2 )3a2 x2a2xa2dxx22c47x2a

8、.48xdx 1c( x2x2a2a2 )349x2a2dx xx2a2a2 ln xx2a 2cx22250x2dx xa2ln xx2a2c( x2a2 )3x251dx 1 arccos acx x 2a2ax52dxx2a2cx2x2a2 xa253x2a 2dx xx 2a2a 2 ln xx2a2c2254( x2a2 )3 dx x (2 x25a2 ) x2a23 a4 ln xx2a 2ca2 dx 18855x x2( x2a2 )3c356x2x2a2 dx x (2 x2a 2 ) x2a2a4ln xx 2a2c8857x2xa2dx x 2a2a arccos ac

9、x58x2a2 dx x2a2ln xx2a2cx2x（八）含有a2x2(a0)的积分59dxx2 arcsin xca2a60dxxc(a2x2 )3a2 a2x2.61a2xdx a2x2cx262xdx 1c(a2x2 )3a2x 263x2dx xa 2x2a 2arcsin xca2x222a64x2dx xx2arcsin xc(a2x2 )3a2a65dx 1 ln aa2x 2cx a 2x2ax66dxa2x2cx2a2x2a2 x67a2x 2dx xa2x2a 2arcsin xc22a68(a 2x2 )3 dx x (5a 22x2 )a2x23 a4 arcsin

10、xc88a69x a2x2 dx 1 ( a2x 2 )3c370x2a 2x2 dx x (2 x2a2 )a2x2a4arcsin xc88a71a2x22x2aa2x2cxdx aa lnxa2x2a2x2x72x2dxxarcsin ac（九）含有ax2bxc ( a0)的积分73dx1ln 2axb2aax2bx ccax2bxca.74ax 2bx cdx 2axbax 2bxc4a4acb228ln 2axb 2 a ax bx c ca375xdx 1ax2bx cax2bx cab2ln 2axb2aaxbxcc767778dx12axbarcsinb2cc bx ax2a4

11、accbxax2 dx 2axbcbxax2b24ac arcsin2axbc4a8 a3b24acxdx 1cbxax2barcsin 2axbccbxax2a2a3b24ac（十）含有xa 或( xa)(bx) 的积分xb79xa dx ( xb)xa(ba)ln( xax b ) cxbxb80xa dx ( xb)xa(ba)arcsinxacbxbxbx8182dx 2arcsinxab)( xa)(bbc (ax)x( xa)(bx)dx 2x ab(x a)( bx)(b a)2arcsinxac44bx( ab)（十一）含有三角函数的积分83sin xdx cos xc.84c

12、os xdx sin xc85tan xdx ln cosxc86cot xdx ln sin xc87secxdx ln tan(4x )c ln secxtan xc288cscxdx ln tan xc ln cscxcot xc289sec2 xdx tan xc90csc2 xdx cot xc91secx tan xdx secxc92cscx cot xdx cscxc93sin2 xdx x1sin 2xc2494cos2 xdx x1sin 2 xc2495sinn xdx 1 sinn 1 x cos xn1sin n2 xdxnn96cosn xdx 1 cosn 1 x

13、 sin xn1cosn 2 xdxnn97dx1cos xn2dxsinnn1sin n 1 xn1sin n2 xx98dx1sin xn2dxcosn1 cosn1 xn1cosn2 xx n99cosm x sinn xdx 1cosm 1 x sin n 1 xm1cosm 2 xsin n xdxmnmnm1cosm 1 x sin n 1 xn1cosm xsin n 2 xdxnmn100sin ax cosbxdx 1cos(ab) x1cos(a b)x c2( a2(ab)b).1011sin(ab)x1sin( ab) xcsin ax sin bxdx 2( a2(

14、ab)b)1021sin(ab) x1sin( ab) xccosax cosbxdx 2( a2( ab)b)dx2a tan xb103ab2arctana22c(a 2b2 )b sin xa2b2dx1a tan xbb2a 21042(a2b2 )ab sin xb2a2ln a tan xbb2a 2c2105adx2bab arctan(ab tan x )c(a 2b2 )b cos xaabab2dx1abtan xab106ln2bac(a22)ab cos xabbabtan xab2ba107adxb2 sin 2 x 1arctan( b tan x)c2 cos2

15、xaba108adxb2 sin 2 x1ln b tan xac2 cos2 x2abb tan xax sin axdx 11c1092 sin axx cosaxa1a22110x2sin axdxx2cosaxx sin axcosaxcaa2a3111x cos axdx 1 cosax1 x sin axca2a112x2 cosaxdx 1 x2 sin ax22 xcosax23 sin axcaaa（十二）含有反三角函数的积分（其中a 0）113arcsin xdx x arcsin xa2x2caa.114x arcsin x dx ( x2a2)arcsinxxa2x2c

16、a24a4115x2 arcsin xdx x3arcsin x1 ( x22a2 )a2x2ca3a9116arccos xdx x arccos xa2x2caa117x arccos xdx ( x2a2)arccos xxa2x2ca24a4118x2 arccos x dx x3arccos x1 ( x22a2 )a 2x2ca3a9119arctan x dx x arctan xa ln( a2x2 )caa2120x arctan x dx 1 (a2x2 )arctan xa xca2a2121x2 arctan x dx x3arctan xa x2a3ln( a2x2

17、)ca3a66（十三）含有指数函数的积分122axdx 1a xcln a123eaxdx 1 eaxca124ax 1(axaxcxe dxa 21)e125xneax dx 1 xn eaxnxn 1eaxdxaa126xa xdx xa x12 a xcln a(ln a)127xna xdx 1xn axnxn 1a xdxln a1ln a128eax sin bxdx a2b2 eax (a sin bxb cosbx)c1129eax cosbxdx a2b2 eax (b sin bxa cosbx)c.130eax sinn bxdx 21 2n2 eax sinn1 bx(

18、a sin bxnb cosbx)abn(n1)b2axsinn2bxdxa2b2n2e131eax cosn bxdx a21 22 eax cosn 1 bx(a cosbxnb sin bx)b nn(n1)b2axn2a2b2n2ecosbxdx（十四）含有对数函数的积分132ln xdx x ln xxc133dx ln ln xcx ln x11134xn ln xdx n1xn 1 (ln x)cn1(ln x) n dx x(ln x) n nn 1dx135(ln x)136xm(ln x)n dx 1xm 1 (ln x) nnxm(ln x)n 1 dxm1m1（十五）含有双曲函数的积分137shxdx chxc138chxdx shxc139thxdx lnch xc140sh2 x