1、(tgx)sec2 x(arcsin x)11x 2(ctgx)csc2 x(arccos x)1(secx)secx tgx1x2(cscx)cscx ctgx(arctgx )1( ax )a x ln a1 x 2(log a x)1(arcctgx )1x ln a1x2tgxdxln cosxCdx2tgx Ccos2 xsec xdxctgxdxln sin xCdxcsc2 xdxctgxCsecxdxln secxtgxCsin 2 xcscxdxln cscxctgxCsecx tgxdxsecxCdx1xCcsc x ctgxdxcsc xCa2x2arctgaa xaax

2、dxCdx1xaln aCx2a22alnashxdxchxCxdx1 ln axCchxdxshxCa2x22aaxdxarcsin xCdxln( xx2a2 )Ca2x2ax2a22sin n xdx2cosnn1 I nI nxdx200nx2a2 dxxx 2a2a2ln( xx2a2 )C22x2a2dxxx2a2a2ln xx2a2C22a2x2 dxxa 2x 2a2arcsin xC22a導數公式：基本積分表：三角函數的有理式積分：（一）含有 axb 的積分 ( a0 )1dx1 ln axbCaxba2(axb) dx a(11)(axb)1C （1）3xd 1( axbb

3、ln axb )C2axbxa4x 2dx 11 (axb) 22b(axb)b2 ln axbCaxba325dxb)1 lnaxbCx(axbxdx1aax bC62( axbxb2 lnxxb)7(axx2 dx 12 (ln axbaxb)Cb)ab8(axx22 dx 13 (axb2b ln axbb2)Cb)aaxb9dxb)2 1b)12ln ax bCx(axb( axbx（二）含有axb 的積分10axbdx 2( axb) 3C3a211xaxbdx 2 (3ax2b)(axb) 3C15a12x2axbdx 2 3 (15a 2x212abx8b2 )( ax b)3C

4、105a13xbdx 22 (ax 2b) ax b Cax3a14x2bdx 2(3a2 x24abx8b2 )axbCax15a31lnaxbbC( b0)dxbaxbb15xaxb2arctanaxbC(b0)bb16dxaxbadxx2 axbx2bxaxbb17axbdx 2axbbxdxxax b18axbdx axbaxdxbx2x2ax（三）含有 x2a 2 的積分19x2dxa2= 1 arctan xCaadxx2n 3dx20( x 2a2 )n = 2( n 1)a2 ( x2a2 )n 12( n 1)a2( x2a 2 )n 121x2dxa 2= 1 ln xaC

5、2axa（四）含有 ax 2b(a0)的積分1 arctana xC(b0)22dxabbax 2b1axblnC (b0)2axbab23xdx 1 ln ax2bCax2b2a24x2dx xbdx2aaax2baxb25dx 1 lnx2bCx(ax 2b)2bax 226dx1adxx2 (ax2b)bxbax2bdxaax2b1C27x3( ax2b)2lnx22bx22b28dxx1dx(ax2b) 2 2b(ax 2b)2bax2b（五）含有 ax 2bxc (a0) 的積分2arctan2axbC(b24ac)29dx4acb24acb2b2ax 2bxc12axb4ac24a

6、c)b2lnbb2C(b4ac 2ax4ac30ax2xdx 1 ln ax2bxcbdxbxc2a2aax2bx c（六）含有x2a 2(a0) 的積分31dx arsh xC1 ln( xx2a2 )Cx2a2a32dxxC( x2x2a2a2 )3a233x2xdx x2a2Ca234xdx 1C( x2x2a2a2 ) 335x2dx xx2a2a2ln( xx2a2 )Cx2a22236x2dx x2xln( xx2a2 )C( x2a 2 )3a237dx 1 lnx2a2aCx x2a2ax38dxx2a2a2 xCx2x2a 239x2a2 dx xx2a2 a2ln( xx2

7、a2 )C2240( x2a2 )3 dx x (2 x25a 2 )x2a2 3 a4 ln( xx2a2 ) C8841 x x2a2 dx 1 ( x2a2 )3C342x2x2a2 dx x (2 x 2a2 )x2a 2a4ln( xx2a2 ) C8843x2xa2 dx x2a2a lnx2a 2aCx44x2a2dx x2a2ln( xx2a2 ) Cx2x（七）含有x2a2(a0)的積分45dx x archxC1 = ln xx2a 2Cx2a2ax46dxxC( x2a2x2a2a2 ) 347x2xa2dx x2a2C48xdx 1C( x2x2a2a2 )349x2a

8、2dx xx2a2a2ln xx2a2Cx22250x2dx xa 2ln xx2a2C( x2a2 )3x251dx 1 arccos aCxx2a2ax52dxx2a2Ca2xx2x2a253x2a2 dx xx2a2a2 ln xx 2a2C2254( x2a2 )3 dx x (2 x25a2 ) x2a 23 a4 ln xx2a2C8855 x x2a2 dx 1 ( x2a 2 )3C356 x2 x2a2 dx x (2 x2a2 ) x2a 2a4 ln xx2a 2C8857x2xa2dx x2a2a arccos aCx58x2a2 dx x2a2ln xx2a2Cx2x

9、（八）含有a2x2(a0) 的積分59dxx2 arcsin xCa2a60dxxC(a2x2x2 ) 3a 2 a261xx2dx a2x2Ca262xdx 1C(a2a2x2x2 )363x2x2dx xa2x2a2arcsin xCa222a64x2dx xx2arcsin xC(a2x2 )3a2a65xdx 1 ln aa2x2Ca2x2ax66dxa2x2Ca2 xx2a2x267a2x2 dx xa2x2a2arcsin xC22a68(a2x2 )3 dx x (5a22x2 )a2x23 a4 arcsin x C88a69 x a2x2 dx 1 (a2x2 ) 3C370

10、x2a2x2 dx x (2 x2a2 )a2x2a4arcsin xC88a71a2x2 dx a2x2a ln aa2x2Cxx72a2x2dx a2x2arcsin xCx2xa（九）含有ax2bxc (a0) 的積分73dx 1 ln 2axb2 aax2bx c Cax 2bxca74ax2bxcdx 2axbax2bxc4a75xdx 1ax2bxcax 2bxca767778dx1 arcsin2axbCc bx ax 2ab24accbxax2dx 2axbbxax2b24ac2axb4ac8 a3arcsinCb24accxdx 1c bxax 22barcsin2axbCb

11、xax 2aa3b24ac（十）含有xa 或( xa)( bx) 的積分xb79xa dx (xb)xa( ba)ln(xaxb )Cxbxb80xa dx (xb)xa(ba)arcsinxaCbxbxbx81dx 2arcsinxa(ab)( xa)( bbCx)x82( xa)(bx)dx 2 xab( x a)(bx)(ba)2arcsinxaC44bx（十一）含有三角函數的積分83sin xdx cos xC84cos xdx sin xC85tan xdx ln cosxC86cot xdx ln sin xC87sec xdx ln tan(x)C ln secxtan xC42

12、88csc xdx ln tan xC ln cscxcot xC289sec2 xdx tanxC90csc2 xdx cot xC91secx tan xdx secxC92csc x cot xdxcscxC93sin 2 xdx x1 sin 2xC2494cos2 xdx x1 sin2xC2495sin n xdx 1 sinn 1 x cosxn1sinn2 xdxnn96cosnxdx 1 cosn1 x sin xn1cosn 2 xdxnn97dx1cosxn2dxsinnxn1sinn 1 xn1sinn2 x98dx1sin xn2dx2 xcosnxn1 cosn1

13、xn1cosn99cosm x sinn xdx 1cosm 1 x sinn 1 xm1cosm2 x sinn xdxmnmn1cosm 1 x sinn 1 xn1cosm x sinn 2 xdxmnmn100 sin ax cosbxdx 1cos(ab) x1cos(ab) x C2( a2( ab)b)101 sin ax sin bxdx 1sin( ab) x1sin(ab)xC2( a2(ab)b)102 cosax cosbxdx 1sin( ab)x1sin(ab) xC2( a2(ab)b)dx2a tan xb103 b2 arctana22C( a2b2)ab s

14、in xa2b2dx1a tan xbb2a2104 ln2C(a 2b2 )ab sin xb2a2a tan xbb2a22105 dx2ab arctan(abtan x ) C ( a2b2 )ab cosxab abab2dx1abtan xab106 ln2baC(a22)aabbatan xabbb cosx2ba107 dx1btan x)Ca2 cos2 xb2 sin2 xabarctan(a108 dxb2 sin2 x1ln b tan xaCa2 cos2 x2abb tan xa109 x sin axdx 12 sin ax1 x cosaxCaa110 x2 s

15、in axdx 1 x2 cosax22x sin ax23 cosaxCx cosaxdx 1a1aa111 cosaxx sin axCa2a112 x2 cosaxdx 1 x2 sin ax22x cosax23 sin axCaaa（十二）含有反三角函數的積分（其中a0）113 arcsin xdx2x2Caxx arcsinaa114 x arcsin x dx ( x2a2)arcsinxxa2x2Ca24a4115 x2 arcsin xdx x3arcsin x1 ( x22a2 )a 2x2Ca3a9arccosxdx22116 xx arccosaxCaa117 x ar

16、ccos xdx ( x2a2)arccos xxa 2x2Ca24a4118 x2arccosxdx x3x122a2)a2x2Ca3arccos( xa9119 arctan x dx arctanxa22)Caxaln( axx 12xa120 arctand22x(a x )arctanxCax2a2121 x2 arctan xdx x3arctan xa x2a3ln( a2x2 )Ca3a66（十三）含有指數函數的積分122 axdx 1a xCln a123 eax dx 1 eaxCa124 xeaxdx 1(ax1)eaxCa2125 xn eaxdx 1 xneaxnxn

17、 1eaxdxaa126 xax d xx1xCxln a a(ln a)2 a127 128 129 130 131 xn a xdx 1 xna xnxn 1axdxln a1ln aeax sin bxdx 22eax ( a sinbxb cosbx)Ca1beax cosbxdx 22 eax (bsin bxa cosbx)Cabeax sin n bxdx 2122 eax sinn 1 bx(a sinbxnbcosbx)ab neax cosn bxdx a2122 eax cosn1 bx(a cosbxnbsin bx)b n（十四）含有對數函數的積分132 ln xdx x ln xxC133 dx ln ln xCx ln x11134 xn ln xdx xn1(ln x) Cn1n1(ln x)n dx x(ln x)nnn 1135 (ln x)dx136 xm (ln x)n dx 1xm 1(ln x)nnxm (ln x)n 1dxm1m1（十五）含有雙曲函數的積分137 shxdx chxC138 chxdx shxC139 thxdx lnch xC140 sh2 xdx x1 sh2xC