1、標(biāo)準(zhǔn)實(shí)用常用積分公式（一）含有 axb 的積分 ( a0 )1dx 1 ln axbCaxba2 ( ax b) dx a(11)(axb) 1C （1 ）3xd1(ax b bln ax b ) Caxbxa24x2dx 11 (axb)22b(axb)b2 ln ax bCaxba325dx1 ln axbCx( axb)bx62dxb)1a2 ln ax bCx (axbxbx7( axxdx 1(ln ax baxb)Cb) 2a2b8x22 dx 13 (ax b 2b ln ax bb2) C(axb)aaxbdx11axb9x(axb)2 b( axb)b2lnxC（二）含有ax

2、b 的積分10axbdx 2(axb)3C3a211xaxbdx 2 (3ax2b)(ax b)3C15a12x2axbdx 2 3 (15a2 x212abx8b2 )(axb)3C105a13xdx 2(ax 2b) ax b Caxb3a2文案大全標(biāo)準(zhǔn)實(shí)用14x222 22axb dx15a3(3a x4abx 8b ) ax b Cdx1 lnaxbbC(b0)15baxbbxaxb2arctanaxbC(b0)bb16dxaxbadxx2bx2bx axbaxb17axb dx 2axbbdxbxxax18axb dx axbadxbx2x2 xax（三）含有 x2a2 的積分19x

3、dx= 1 arctan xC2a2aadxx2n 3dx20( x2a2 )n = 2( n 1)a2 ( x2a2 )n 12( n 1)a2( x2a 2 )n 121xdx= 1 lnxaC2a 22axa（四）含有 ax2b(a0) 的積分1arctana xC( b0)22dxabbax2baxb1lnC(b0)2abaxb23xdx 1 ln ax2bCax2b2a24x2dx xbdxbax2baaax2文案大全標(biāo)準(zhǔn)實(shí)用25dx 1 lnx2bCx(ax 2b) 2bax226dx1adxx2 (ax2bxbax2bb)27dx aax2b1xlnx2C3(ax2b)2b22b

4、x228dxx1dx(ax 2b) 2 2b( ax2b)2bax2b（五）含有 ax2bxc ( a0) 的積分2b2arctan2axbC(b24ac )29dx4ac4acb222axbxc12axbb4acC (b2ln4ac )b24ac2axbb24ac30xdx 1ln ax2bxcbdxcax2bxc2a2aax 2bx（六）含有x2a2(a0)的積分31dxa2 arsh xC1 ln( xx2a2 )Cx2a32dxxC(x 2a2 x2a2a2 ) 333xa2dx x2a2Cx234xdx 1C(x2x2a2a2 ) 335x2a2dx xx2a 2a2ln( xx2a

5、2 )Cx222文案大全標(biāo)準(zhǔn)實(shí)用36x2a 2 )3dx xln( xx2a2 )C( x2x2a237dx 1 lnx2a2aCx x2a2ax38dxx2a2Cx2x2a2a2 x39x2a2 dx xx2a2a2ln( xx2a2 )C2240( x2a2 )3 dx x(2 x25a2 )x2a23a4 ln( xx2a2 ) C8841x x2a2 dx 1 (x2a2 ) 3C342x2 x2a2 dx x (2 x843x2a2 dx x2ax2a2 )x2a2a4ln( xx2a2 ) C82a lnx2a2aCx44x2a2dx x2a2ln( xx2a2 )Cx2x（七）含

6、有x2a2(a0)的積分45dx x archxC1 =ln xx2a2Cx2a2xa46dxxC( x2a2x2a2 ) 3a247xa2dx x2a 2Cx248xdx 1C(x2x2a2 )3a2文案大全標(biāo)準(zhǔn)實(shí)用49x2dx xx2a2a2ln xx2a2Cx2a22250x2xa2ln xx2a2C(x2a2 )3 dxx251dx1aCx2arccosxa 2 ax52dxx2a2Cx2x2a2 xa253x2a2 dx xx2a2a2ln xx2a2C2254( x2a2 )3 dx x (2 x25a2 ) x2a23 a4 ln xx2a2C8855 x x2a2 dx 1(x

7、2a2 )3C356x2 x2a2 dx x (2 x2a2 ) x2a 2a4 ln xx2a2C8857x2a2dx x2a2a arccos aCxx58x2a2 dx x2a2ln xx2a2Cx2x（八）含有a2x2(a0) 的積分59dxx2 arcsin xCa2a60dxxC(a 2a2 a2x2 ) 3x261xx2dx a2x2Ca2文案大全標(biāo)準(zhǔn)實(shí)用62xdx 1C(a2x2 )3a2x263x2x2dx xa2x2a2arcsin xCa222a64x2dx xx2arcsin xC(a2x2 )3a 2a65dxx2 1 ln aa 2x2Cxa2ax66dxa2x2C

8、a 2x2a2 xx267a 2x2 dx xa2x2a2arcsin xC22a68(a2x2 )3 dx x (5a22x2 )a2x23a4 arcsin xC88a69 x a 2x2 dx 1(a2x2 )3C370 x2a2x2dx x2a2)a2x2a4xC(2 x8arcsin8a71a2xx2 dx a2x2a ln aa2x2Cx72a2x2dx a 2xx2arcsin xCx2a（九）含有ax2bxc (a0) 的積分73ax2dx 1ln 2axb2aax 2bx cCbxca74ax 2bxcdx 2axbax2bxc4a文案大全標(biāo)準(zhǔn)實(shí)用4acb2ln 2axb2a

9、ax2bxc C8a375ax2xdx 1ax2bxcbxcabln 2axb2aax2bx cC2 a376dx12axbCcbxax 2arcsinb24aca77cbxax2 dx 2axbcbxax2b24ac arcsin2axbC4a8a3b24ac78xdx 1cbxax2barcsin2axbCc bx ax 2a2 a3b24ac（十）含有xa 或( xa)( bx) 的積分xb79xa d (xb)xa(ba)ln(xaxb)Cxxxbb80xa d(xb)xa(ba)arcsinxaCbxbxbxx81dx 2arcsinxaC( ab)(x a)( bbxx)82( x

10、a)(bx)dx 2 xab( xa)( bx)(ba)2arcsinxaC44bx(ab)（十一）含有三角函數(shù)的積分83sin xdx cos xC文案大全標(biāo)準(zhǔn)實(shí)用84 cos xdx sin xC85tan xdx ln cosxC86cot xdx ln sinxC87 sec xdx ln tan(4x )C ln secxtan x C288 csc xdx ln tan xC ln cscxcot xC289 sec2 xdx tan xC90 csc2 xdx cotxC91 secx tan xdx secxC92 csc x cot xdx cscxC93 sin2 xdx

11、x1sin 2xC2494 cos2 xdx x1sin 2xC2495sinn d 1n1n 1sinn2xdxx xsinxcos xnn96 cosn xdx 1 cosn 1 x sin xn1cosnnn97dx1cos xn2dxsin n xn1sinn1 xn1sinn298dx1sin xn2dxcosn xn 1cosn 1 xn1cosn299 cosm xsin n xdx 1cosm 1 x sinn 1 xmn2 xdxxxm1cosm 2 x sinn xdxmn1cosm 1 x sinn 1 xn1cosm x sin n2 xdxmnmn100 sin ax

12、 cosbxdx 11cos(a b) xC2( acos(a b) x2( ab)b)文案大全標(biāo)準(zhǔn)實(shí)用101sin ax sin bxdx 1sin( ab) x1b)xC2( a2(asin(ab)b)102cosax cosbxdx 1sin( ab)x1b) xC2( a2(asin( ab)b)dx2a tan xb103ab2arctana22C( a2b2 )bsin xa2b2dx1a tan xbb2a2( a2b2 )1042absin xb2a 2 ln a tan xbb2a2C2105adxa2ab arctan(ab tan x )C( a2b2 )bcos xba

13、bab2dx1abtan xab106ln2baC(a22)abcos xabbatan xabb2ba107dx 1arctan( b tan x)Ca2 cos2 xb2 sin2 xaba108dx 1ln b tan xaCa2 cos2 xb2 sin2 x2abb tan xa109xsin axdx 11Ca2 sin axx cosaxa110x2 sin axdx 1 x2 cosax2 x sin ax2cosaxCaa2a3111x cosaxdx 1cosax1x sin axCa2a112x2 cosaxdx 1 x2 sin ax22x cosax23 sin ax

14、Caaa0 ）a（十二）含有反三角函數(shù)的積分（其中113 arcsin xdxa2x2Cxx arcsinaa文案大全標(biāo)準(zhǔn)實(shí)用114xarcsinxd x2a2xx22()arcsinaxCax24a4115x2 arcsin xdx x3arcsin x1 ( x22a2 )a2x2Ca3a9116 arccos xd x arccosxa2x2Caxa117xarccosxd x2a2xx22()arccosaxCax24a4118x2 arccos x dx x3arccos x1 ( x22a2 )a2x2Ca3a9119arctan x dx x arctan xa ln( a2x2

15、 )Caa2120xarctan x d1(a22xaxCax2x)arctan2a121x2 arctan x dx x3arctan xa x2a3ln( a2x2 )Ca3a66（十三）含有指數(shù)函數(shù)的積分122axd 1xCxaln a123eax dx 1 eaxCa124eaxd 1axCxxa2 (ax1)e125xn eaxdx 1 xneaxnxn 1eaxdxaa126xa xdx xax12 axCln a(ln a)127xn a xdx 1xna xnxn 1a xdxln a1ln a128 eax sin bxdx a2b2 eax ( a sin bxb cosb

16、x )C1129 eax cosbxdx a2b2 eax (bsin bxa cosbx)C文案大全標(biāo)準(zhǔn)實(shí)用130eax sin n bxdx 212n2 eax sin n 1 bx( a sin bxnb cosbx )abn( n1)b2axsinn2bxdxa2b2n2e131 eax cosnd212axcosn1bx(a cosbxnbsin bx)bx xabn2 en( n1)b2axn2a2b2n2ecosbxdx（十四）含有對數(shù)函數(shù)的積分132 ln xdx x ln xxC133dx ln ln xCxln x11134xn lndxn 1(ln x)Cx xn1n 1135 (ln x)n dx x(ln x)nn1n (ln x)dx136xm (ln x)n dx 1xm 1(ln x)nnxm (ln x)n 1 dxm1m1（十五）含有雙曲函數(shù)的積分137 shxdx chxC138 chxdx shxC139thxdx lnchxC140 sh2 xdx x1 sh2xC24141 ch 2 xdx x1sh2xC24（十