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)3C105a13xdx 22 (ax2b)axbCaxb3a14x2dx 2(3a2 x24abx 8b2 ) ax b Caxb15a31axbbC(b0)dxlnaxbb15bx axb2arctanaxbC(b0)bb16dxaxbadxx2axbbx2bx axb17axbdx 2axbbxdxbxax18axbdx axbaxdxbx2x2ax（三）含有 x2a2的积分19dx=1xCx2a2aarctanadxx2n 3dx20( x2a2 )n = 2( n 1)a 2( x2a2 )

5、 n 12( n 1) a2( x2a2 )n 121dx=1 lnxaCx2a22axa（四）含有 ax2b( a0) 的积分1arctana xC( b0)22dxabbax2b1axblnC(b0)2abaxb23xdx 1ln ax2bCax2b2ax2xbdx242dx a2baxbaax25dx 1 lnx2bCx( ax2b)2bax226dx1adxbxbax 2bx2 (ax2b)27dxalnax2b1Cx2x3 (ax 2b)2b22bx228dxx1dx2b(ax 2b)2bax 2b(ax2b)2（五）含有 ax2bxc (a0) 的积分2arctan2axbC( b

6、24ac)29dx4acb24acb22bx12axbb24acaxcln(b24ac)b22axbb2C4ac4ac30xdx 1 ln ax2bxcbax 2dxax2bxc2a2abx c（六）含有x2a2(a0) 的积分31dxa2 arsh xC1 ln( xx2a2 ) Cx2a32dxxCa 2 x2a2( x2a2 )333xa2dx x2a2Cx234xdx 1Cx2a2( x2a 2 )335x2dx xx2a2a2ln( xx2a2)Cx2a22236x2xa2ln( xx2a2 )C( x2a2 )3 dxx237dx 1 lnx 2a2aCx x 2a2ax38dxx

7、2a2Cx2x2a2 xa239x2a2dx x2a2a2ln( xx2a2)Cx2240( x2a2 )3 dx x(2 x25a2 )x2a23a4 ln( xx2a2 ) C8841x x2a2 dx 1( x2a2 )3C342x2x 2a2 dx x (2 x2a 2 )x2a2a4ln( xx 2a2 ) C8843x2xa2 dx x2a 2a lnx2a2aCx44x2a2dx x2a2ln( xx2a2 )Cx2x（七）含有x2a2(a0) 的积分45dx x archxC1 = ln xx2a 2Cx2a2xa46dxxC( x2a2 x2a2a 2 )347x2xa2dx

8、 x2a2C48xdx 1C( x2x2a2a2 )349x2a2dx xx2a2a2 ln xx2a 2Cx22250x2dx xln xx2a2C( x2x2a2a2 )351dxa2 1 arccos aCx x 2ax52dxx2a2Cx2x2a2 xa253x2a2 dx xx 2a2a 2 ln xx2a2C2254( x2a2 )3 dx x (2 x25a2 ) x2a23 a4 ln xx2a 2Ca2 dx 18855x x2( x2a2 )3C356x2x2a2dxx2a2) x2a2a4ln xx2a2C(2 x8857x2xa2dx x 2a2a arccos aCx

9、58x2a2 dx x2a2ln xx2a2Cx2x（八）含有a2x2(a0)的积分59dxx2 arcsin xCa2a60dxxC(a2x2x2 )3a2 a261a2xdx a2x2Cx262xdx 1C(a2x2 )3a2x263x2dx xa 2x2a 2arcsin xCa2x222a64x2dx xx2arcsin xC(a2x2 )3a2a65dx 1 ln aa2x2Cx a 2x2ax66dxa2x2Cx2a2x2a2 x67a2x2 dx xa2x2a 2arcsin xC22a68(a 2x2 )3 dx x (5a 22x2 )a2x23 a4 arcsin xC88

10、a69x a2x2 dx 1 ( a2x 2 )3C370x2a 2x2 dx x (2 x2a2 )a2x2a4arcsin xC88a71a2xx2 dx a2x2a ln aa2x2Cxa2x2a2x2x72x2dxxarcsin aC（九）含有ax2bxc ( a0)的积分73dx 1ln 2axb2aax2bx cCax2bxca74ax 2bx cdx 2axbax 2bxc4a4acb228ln 2axb 2 a ax bx c Ca375xdx 1ax 2bx cax2bx cabln 2ax b 2 a ax 2bx c C2 a3767778dx1 arcsin2axbCc

11、 bx ax2ab24accbxax2 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

12、 cos xC84cos 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

13、1 cosn 1 x 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 bxd

14、x 2( a2( 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

15、)C2 cos2 xaba108adxb2 sin 2 x1ln b tan xaC2 cos2 x2abb tan xax sin axdx 11C1092 sin axx cosaxa1a22110x2sin axdxx2cosaxx sin axcosaxCaa2a3111x cosaxdx 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 xa2x2Caa114x arcsin x dx ( x2a2)arcsin

16、xxa2x2Ca24a4115x2 arcsin xdx x3arcsin x1 ( x22a2 )a2x2Ca3a9116arccos xdx x arccos xa2x2Caa117x arccosxdx ( 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(

17、 a2x2 )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)C130eax sinn bxdx 21 2n2 eax sinn

18、1 bx(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 xC140