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 xdxt

2、gx 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)1dx

3、 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 (3ax2b)

4、(axb)3C15a12x2axbdx 2 3 (15a2 x212abx8b2 )(axb)3C105a13xx22 (ax2b)axbCdaxb3a欢迎下载214x2dx 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( x2a

5、2 ) 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欢迎下载3x2xbdx242dx 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2arctan2a

6、xbC( b24ac)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欢迎下载435x2xx2a2a2ln( xx2a2)Cx2a2dx 2236x2dx xa2ln( xx2a2 )C( x2a2 )3x237xdx 1 lnx2a2aCx

7、2a2ax38dxx2a2Cx2x2a2a2 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 x2a2xa2d

8、xx22C47x2a欢迎下载548xdx 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

9、2a2a arccos aCx58x2a2 dx x2a2ln xx2a2Cx2x（八）含有a2x2(a0)的积分59dxx2 arcsin xCa2a60dxxC(a2x2 )3a2 a2x2欢迎下载661a2xdx 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

10、 )a2x23 a4 arcsin 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欢迎下载774ax 2bx cdx 2axbax 2bxc4a4acb228ln 2axb 2 a ax bx c Ca375xdx 1ax2bx cax2bx cab2ln 2axb2aaxbxcC767778dx12

11、axbarcsinb2Cc bx ax2a4accbxax2 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

12、积分欢迎下载883sin xdx cos xC84cos xdx sin x C85tan xdx ln cosxC86cot xdx ln sin xC87secxdx ln tan(x ) C ln secxtan x C4288cscxdx ln tan xC ln cscx cot xC289sec2 xdx tan xC90csc2 xdx cot xC91secx tan xdx secxC92cscx cot xdx cscxC93949596979899sin2 xdx x1sin 2xC24cos2 xdx x1 sin 2 xC24sinn xdx 1 sinn 1 x c

13、os xn1 sin n2 xdxnncosn xdx 1 cosn 1 x sin xn1cosn 2 xdxnndx1cos xn2dxsinn x1sin n 1 xn1sin n2 xndx1sin xn2dxcosn x1 cosn1 xn1cosn2 xncosm x sinn xdx 1cosm 1 x sin n 1 xm1cosm 2 xsin n xdxmnmnm1cosm 1 x sin n 1 xn1cosm xsin n 2 xdxnmn欢迎下载9100sin ax cosbxdx 1cos(ab) x1cos(a b)x C2( a b)2(ab)101sin ax

14、 sin bxdx 1sin(ab)x1sin( ab) xC2( a2( ab)b)102cosax cosbxdx 1sin(ab) x1sin( ab) xC2( ab)2( ab)dx2a tan xb103ab2arctana22C(a 2b2 )b sin xa2b2dx1a tan xbb2a 222104ln2C(a)bab sin xb2a2a tan xbb2a 22105dx2abarctan(abxC22ab cos xbabatan)(ab )ab2dx1abtan xab106ln2baC(a2b2)ab cos xabbatan xab2ba107dx1btan

15、x)Ca2 cos2 xb2 sin 2 xarctan(aba108adxb2 sin 2 x1ln b tan xaC2 cos2 x2abb tan xa11C109x sin axdx 2 sin axx cosaxa1a22110x2sin axdxx2cosaxCaa2x sin ax3 cosaxa111x cos axdx 12cosax1 x sin axCaa112x2cosaxdx 1x2sin ax2xcosax2Caa2a3 sin ax（十二）含有反三角函数的积分（其中a0）欢迎下载10113xxa22Carcsin dx x arcsinaxa114x arcsi

16、n x dx ( x2a2)arcsinxxa2x2Ca24a4115x2 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 d

17、x x3arctan xa x2a3ln( a2x2 )Ca3a66（十三）含有指数函数的积分122123124125126axdx 1 a xCln aeaxdx 1 eaxCaxeax dx 12 (ax1)eaxCaxneax dx 1 xn eaxnxn 1eaxdxaaxa xdx x a x12 a xCln a(ln a)127xna xdx 1xn axnxn 1a xdxln a1ln a128eax sin bxdx a2b2 eax (a sin bx b cosbx) C欢迎下载11129eax cosbxdx a21b2 eax (b sin bxa cosbx)C1

18、30eax sinn bxdx a212n2 eax sinn 1 bx(a sin bxnb cosbx)bn(n1)b2axsinn2bxdxa2b2n2e131eax cosn bxdx 1axn 1bx(a cosbxnb sin bx)a2b2 n2 ecosn(n1)b2eaxn2bxdxa2b2n2cos（十四）含有对数函数的积分132ln xdx x ln xx C133dxC ln ln x134135136x ln x11xn ln xdx xn 1 (ln xn1n1(ln x) n dx x(ln x) nn 1n (ln x)xm(ln x)n dx 1xm 1 (ln x) nm1)Cdxn xm(ln x)n 1 dxm 1（十五）含有双曲函数的积分137shxdx chxC138chxdx shxC139thxdx lnch xC140sh2 xdx x1