不定积分表

常 用 积 分 公 式

（一）含有axb的积分(a0) 1．

dx1＝axbalnaxbC

2．(axb)dx＝

1(axb)1C（1）

a(1)3．

x1dx(axbblnaxb)C ＝axba2x211dx＝3(axb)22b(axb)b2lnaxbC 4．axba25．

dx1axb＝x(axb)blnxC

6．

dx1aaxb＝lnC 22x(axb)bxbx7．

1bx(lnaxb)C dx＝(axb)2a2axb1b2x2)C 8．dx＝3(axb2blnaxbaaxb(axb)29．

dx11axb＝lnC

x(axb)2b(axb)b2x（二）含有axb的积分

23(axb)C 3a2(3ax2b)(axb)3C 11．xaxbdx＝215a22(15a2x212abx8b2)(axb)3C 12．xaxbdx＝3105a10．

axbdx＝13．

2xdx＝2(ax2b)axbC

3aaxb

14．

2x2(3a2x24abx8b2)axbC dx＝315aaxbdx15．＝xaxb16．

1lnbaxbbC(b0)axbb

2axbarctanC(b0)bbx2dxaxbadx＝ bx2bxaxbaxb17．

dxaxb dx＝2axbbxxaxb18．

axbaxbadx＝ dxx2x2xaxb22（三）含有xa的积分 19．

dx1xarctanC =x2a2aa20．

x2n3dxdx=(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n1

21．

dx1xa=lnC

x2a22axa2（四）含有axb(a0)的积分

1arctandxab22．2＝axb1ln2ab23．

axCb(b0)

axbC(b0)axbx12dxlnaxbC ＝ax2b2a

．x224axbdx＝xbdx2aaax2b 25．dx1x2x(ax2b)＝

2blnax2bC 26．

dx1adx2(ax2b)＝bxxbax2b

27．dxaax2b1x3(ax2b)＝2b2lnx22bx2C 28．

dxx(ax2b)2＝2b(ax2b)1dx2bax2b

（五）含有ax2bxc(a0)的积分

22ax2arctanb29．dx4acb4acb2Cax2bxc＝12axbb2ln4acCb24ac2axbb24ac30．

x12bdxax2bxcdx＝2alnaxbxc2aax2bxc（六）含有x2a2(a0)的积分 31．

dxxx2a2＝arshC1＝ln(xx2a2a)C 32．

dxx(x2a2)3＝a2x2a2C

33．

xx2a2dx＝x2a2C

34．

x1(x2a2＝)3dxx2a2C

(b24ac)(b24ac)

35．

x2a22xaln(xx2a2)C dx＝22x2a2x236．

x2(x2a2)3dx＝xx2a2ln(xx2a2)C

1x2a2aC 37．＝ln22axxxadx38．

x2x2a2＝C 222axxadx2239．

x2a22xaln(xx2a2)C xadx＝22x342222223(2x5a)xaaln(xx2a2)C (xa)dx＝88122(x2a2)3C 41．xxadx＝340．42．x2xa42222xadx＝(2xa)xaln(xx2a2)C

882243．

x2a2ax2a222C dx＝xaalnxx44．

x2a2x2a2dx＝ln(xx2a2)C 2xx（七）含有x2a2(a0)的积分 45．

dxx2a2＝

xxarchC1=lnxx2a2C xa＝46．

dx(xa)xx2a2223xa2xa22C

47．

dx＝x2a2C

48．

x(xa)x2223dx＝1xa22C

49．

x2a22xalnxx2a2C dx＝22x2a2x2(x2a2)3dx＝50．

xx2a2lnxx2a2C

51．

xx2dxx2a2dx＝1aarccosC ax52．

x2a2＝C 222axxa2253．

x2a22xalnxx2a2C xadx＝22x34222222223(2x5a)xaalnxxaC (xa)dx＝88122(x2a2)3C 55．xxadx＝354．56．x2xa42222xadx＝(2xa)xalnxx2a2C

882257．

ax2a2dx＝x2a2aarccosC

xxx2a2x2a222＝dxlnxxaC 2xx58．

（八）含有a2x2(a0)的积分 59．

dxa2x2＝arcsinxC a60．

dx(ax)223＝xa2ax22C

61．

xa2x2dx＝a2x2C

1ax2262．

x(ax)x2223dx＝C

63．

x2a2x2axarcsinC dx＝22aa2x2x2(a2x2)3dx＝64．

xa2x2arcsinxC a1aa2x2C 65．＝ln22axxaxdx66．

x2a2x2＝C 222axaxdx2267．

x2a2x2axarcsinC axdx＝22ax34x2222223(5a2x)axaarcsinC (ax)dx＝88a122(a2x2)3C 69．xaxdx＝368．70．x2xa4x2222axdx＝(2xa)axarcsinC

88a2271．

aa2x2a2x222C dx＝axalnxx72．

a2x2a2x2x＝dxarcsinC 2xxa（九）含有ax2bxc(a0)的积分 73．

dxax2bxc＝1ln2axb2aax2bxcC a

74．

ax2bxcdx＝2axbax2bxc 4a

xax2bxc24acb8a32ln2axb2aaxbxc C75．

dx＝1ax2bxc a

dxcbxax22b2a3ln2axb2a2axbxc C76．

＝12axbarcsinC

2ab4ac77．

2axbb24ac2axb2cbxaxdx＝cbxaxarcsinC

324a8ab4acxcbxax2dx＝1b2axbcbxax2arcsinC

32a2ab4ac78．

（十）含有xa或(xa)(bx)的积分 xbxb)C

79．

xaxadx＝(xb)(ba)ln(xaxbxb80．

xaxaxadx＝(xb)(ba)arcsinC bxbxbxdxxa＝2arcsinCbx(xa)(bx)(ab)

81．

82．

2xab(ba)2xa(xa)(bx)dx＝(xa)(bx)arcsinC

44bx (ab) （十一）含有三角函数的积分 83．sinxdx＝cosxC



84．cosxdx＝sinxC 85．tanxdx＝lncosxC 86．cotxdx＝lnsinxC 87．secxdx＝lntan(x)C＝lnsecxtanxC 4288．cscxdx＝lntanxC＝lncscxcotxC 289．secxdx＝tanxC 90．cscxdx＝cotxC 91．secxtanxdx＝secxC 92．cscxcotxdx＝cscxC

22x1sin2xC 24x1294．cosxdx＝sin2xC

241n1n1n2nsinxdx 95．sinxdx＝sinxcosxnn1n1n1n2ncosxdx 96．cosxdx＝cosxsinxnndx1cosxn2dx97．＝

sinnxn1sinn1xn1sinn2xdx1sinxn2dx98．＝ nn1n2cosxn1cosxn1cosx1m1m2nmncosm1xsinn1xcosxsinxdx 99．cosxsinxdx＝

mnmn1n1cosm1xsinn1xcosmxsinn2xdx ＝mnmn93．sinxdx＝

2100．sinaxcosbxdx＝11cos(ab)xcos(ab)xC

2(ab)2(ab)

101．sinaxsinbxdx＝11sin(ab)xsin(ab)xC

2(ab)2(ab)102．cosaxcosbxdx＝

11sin(ab)xsin(ab)xC

2(ab)2(ab)atanxb2C22ab103．

2dx＝absinxa2b2arctan(a2b2)

x22bbadx12104．＝lnC22xabsinxbaatanbb2a22atan105．

(a2b2)

dx2ababx＝arctan(tan)Cabcosxababab2(a2b2)

xdx1ab2106．＝lnabcosxabbaxtan2tan107．

abbaCabba(a2b2)

dx1barctan(tanx)C ＝a2cos2xb2sin2xabadx1btanxa＝lna2cos2xb2sin2x2abbtanxaC

108．

11sinaxxcosaxC a2a12222110．xsinaxdx＝xcosax2xsinax3cosaxC

aaa11111．xcosaxdx＝2cosaxxsinaxC

aa12222112．xcosaxdx＝xsinax2xcosax3sinaxC

aaa（十二）含有反三角函数的积分（其中a0）

xx22113．arcsindx＝xarcsinaxC

aa109．xsinaxdx＝

xx2a2xx2ax2C 114．xarcsindx＝()arcsina24a4xx3x12222115．xarcsindx＝arcsin(x2a)axC

a3a92116．arccosdx＝xarccosxaxa2x2C axx2a2xx2ax2C 117．xarccosdx＝()arccosa24a4xx3x12222118．xarccosdx＝arccos(x2a)axC

a3a92xxadxxarctanln(a2x2)C ＝aa2x12xa2120．xarctandx＝(ax)arctanxC

a2a2119．arctanxx3xa2a3ln(a2x2)C 121．xarctandx＝arctanxa3a662（十三）含有指数函数的积分

1xaC lna1axax123．edx＝eC

a1axax124．xedx＝2(ax1)eC

a1naxnn1axnax125．xedx＝xexedx

aa122．adx＝

x126．xadx＝

nxxxx1aaxC 2lna(lna)1nxnn1xxaxadx lnalna1axeax(asinbxbcosbx)C 128．esinbxdx＝22ab1axaxe(bsinbxacosbx)C 129．ecosbxdx＝22ab127．xadx＝

130．esinbxdx＝

axn1eaxsinn1bx(asinbxnbcosbx) 222abnn(n1)b2axn2esinbxdx 2ab2n2131．ecosbxdx＝

axn1axn1ecosbx(acosbxnbsinbx) 222abnn(n1)b2axn2ecosbxdx 222abn（十四）含有对数函数的积分 132．lnxdx＝xlnxxC

dxxlnx＝lnlnxC

1n11nx(lnx)C 134．xlnxdx＝

n1n1133．

135．(lnx)dx＝x(lnx)n(lnx)136．x(lnx)dx＝

nnn1dx

mn1nmn1xm1(lnx)nx(lnx)dx m1m1（十五）含有双曲函数的积分 137．shxdx＝chxC 138．chxdx＝shxC 139．thxdx＝lnchxC

x1sh2xC 24x12141．chxdx＝sh2xC

24140．shxdx＝2（十六）定积分 142．143．

cosnxdx＝sinnxdx＝0

cosmxsinnxdx＝0

144．

0,mncosmxcosnxdx＝ ,mn

0,mn145．sinmxsinnxdx＝

,mn0,mn146．sinmxsinnxdx＝cosmxcosnxdx＝

00,mn2147． In＝ In＝

20sinxdx＝cosnxdx

n20n1In2 nn1n342 （n为大于1的正奇数） In，I1＝1 nn253n1n331In（n为正偶数），I0＝

nn24222