(整理)常用积分公式

常 用 积 分 公 式

（一）含有axb的积分(a0)

dx1lnaxbCaxba1．＝

11(axb)C(axb)dxa(1)2．＝（1）

x1dx(axbblnaxb)C23．axb＝a

11x222(axb)2b(axb)blnaxbCdx3a24．axb＝

dx1axbx(axb)blnxC5．＝ dx1aaxblnCx2(axb)bxb2x6．＝ x1bdx(lnaxb)C2(axb)2aaxb7．＝

2x21b(axb)2dxa3(axb2blnaxbaxb)C8．＝

dx11axblnCx(axb)2b(axb)b2x9．＝

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（二）含有axb的积分

2(axb)3Caxbdx＝3a

10．2(3ax2b)(axb)3Cxaxbdx211．＝15a

12．x22(15a2x212abx8b2)(axb)3Caxbdx3＝105a

13．

x2dx(ax2b)axbCaxb＝3a2

x22dx(3a2x24abx8b2)axbC3axb＝15a

14．

dxxaxb15．＝1lnbaxbbC(b0)axbb2axbarctanC(b0)bb

16．

x2dxaxbadxbx2bxaxb axb＝

17．

dxaxb2axbbdxxaxbx＝ 18．

axbadxaxbdxx2xaxb x2＝

22（三）含有xa的积分

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dx1xarctanC22xaaa19．=

dxx2n3dx(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n120．= 1xadxlnC2221．xa=2axa

2axb(a0)的积分 （四）含有

1arctanab1lndx2ab2axb22．＝axCb(b0)axbC(b0)axb

x1dxlnax2bC223．axb＝2a

x2xbdxdx2224．axb＝aaaxb

1x2dxx(ax2b)2blnax2bC25．＝

dx1adxx2(ax2b)bxbax2b26．＝

ax2bdxa1lnC32x(axb)2b222x2bx27．＝

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28．dxx1dx(ax2b)2＝2b(ax2b)2bax2b

（五）含有ax2bxc(a0)的积分

22ax4acb2arctanb4acb2Cdx12axbb2ln4ac29．ax2bxc＝b24ac2axbb24acCx30．ax2bxcdx1＝2alnax2bxcbdx2aax2bxc（六）含有x2a2(a0)的积分

x31．

dxx2a2＝

arshaC1＝ln(xx2a2)C xx32．

d(x2a2)3＝a2x2a2C

xx2a2dx33．

＝x2a2C

x134．

(x2a2)3dxx2a2C＝

x2x2a2dxx＝2x2a2a235．

2ln(xx2a2)C

x22(x2a2)3dxx＝

x2a2ln(xxa2)C36．

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(b24ac)(b24ac)

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37．

xxdxx2a2＝dx1x2a2alnCax

38．

2x2a2C222xa＝ax

39．2xa22xaln(xx2a2)Cx2a2dx2＝2

40．(x2a2)3dxx3(2x25a2)x2a2a4ln(xx2a2)C8＝8

1(x2a2)3Cxxadx41．＝3

2242．

2x4xa2222(2xa)xaln(xx2a2)Cx2a2dx8＝8

43．

x2a2a22x2a2xaalnCdxxx＝

x2a2x2a222dxln(xxa)C2xx＝

44．

22(a0)的积分 xa（七）含有45．

xxarchC1lnxx2a2Cax2a2＝x=

dxdx(x2a2)346．

＝ax2xa22C

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47．

xxa22dx22xaC ＝48．

x(x2a2)3dx1xa22C＝

49．

xxx2x2a2x2dxx2a22xalnxx2a2C2＝2

xx2a250．

(xa)223dx＝

lnxx2a2C

dxx2a2＝dx51．

1aarccosCax

52．

2x2a2Cx2a2＝a2x

53．2xa22xalnxx2a2Cx2a2dx2＝2

．x342222(2x5a)xaalnxx2a2C(xa)dx8＝8

22355．

22xxadx1(x2a2)3C＝3

x56．24xa22222222(2xa)xalnxxaCxadx8＝8

57．

a22x2a2xaaarccosCdxxx＝

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58．

x2a2x2a2dxlnxx2a2C2xx＝

22（八）含有ax(a0)的积分

59．

dxa2x2＝

arcsinxCa x60．

dx(ax)223＝a2ax22C

xax22dx61．

22＝axC

62．

x(a2x2)3dx1＝ax22C

63．

xx2xax22dxaxarcsinCa2x222a＝

x2x2(a2x2)3dxx22ax＝arcsin．

xCa

dxa2x2＝dx65．

1aa2x2lnCax

66．

2a2x2Ca2x2＝a2x

67．2xax2222axarcsinCaxdx2a＝2

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68．x34x2222(5a2x)axaarcsinC(ax)dx88a＝

2231(a2x2)3Cxaxdx69．＝3

2270．

2x4xax222222(2xa)axarcsinCaxdx88a＝

71．

aa2x222a2x2axalnCdxxx＝

a2x2a2x2xdxarcsinCx2xa＝

72．

2axbxc(a0)的积分 （九）含有73．

12ln2axb2aaxbxcC2aaxbxc＝ 2axbax2bxcaxbxcdx＝4a 2dx74．4acb28a3ln2axb2aax2bxcC

75．

xax2bxcdx1ax2bxc＝a b2a3ln2axb2aax2bxcC

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76．

dxcbxax2＝

12axbarcsinC2ab4ac

77．2axbb24ac2axb2cbxaxarcsinC2cbxaxdx324a8ab4ac＝

xcbxax21b2axbcbxax2arcsinC32a2ab4ac＝ 78．

dx（十）含有xaxb或(xa)(bx)的积分

79．

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

80．

xaxaxadx(xb)(ba)arcsinCbx＝bxbx dx(xa)(bx)xaCbx2arcsin81．＝

(ab)

82．2xab(ba)2xa(xa)(bx)arcsinC(xa)(bx)dx44bx＝

(ab)

（十一）含有三角函数的积分

sinxdx83．＝cosxC cosxdx84．＝sinxC tanxdxlncosxC85．＝

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86．cotxdx＝

lnsinxC

xlntan()CsecxdxlnsecxtanxC4287．＝＝

cscxdx88．＝

2secxdxlntanxClncscxcotxC2＝

．

＝tanxC

90．

2cscxdx＝cotxC

secxtanxdx91．＝secxC cscxcotxdx92．＝cscxC

x1sin2xC＝24

93．

2sinxdxx1sin2xCcosxdx94．＝24

21n1n1sinxcosxsinn2xdxsinxdxn95．＝n

n1n1n1cosxsinxcosn2xdxcosxdxn96．＝n

ndx1cosxn2dxn1nn297．sinx＝n1sinxn1sinx

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dx1sinxn2dxn1nn298．cosx＝n1cosxn1cosx

1m1m1n1cosxsinxcosm2xsinnxdxcosxsinxdxmn99．＝mn

mn1n1cosm1xsinn1xcosmxsinn2xdxmn＝mn 11cos(ab)xcos(ab)xCsinaxcosbxdx2(ab)2(ab)100．＝

11sin(ab)xsin(ab)xCsinaxsinbxdx2(ab)2(ab)101．＝

11sin(ab)xsin(ab)xCcosaxcosbxdx2(ab)2(ab)102．＝

2dxarctan22103．absinx＝abatanxb2C22ab(a2b2)

x22bba12lnC22xdxbaatanbb2a22104．absinx＝

atan(a2b2)

2ababxdxarctan(tan)Cab2105．abcosx＝abab(a2b2)

x1ab2lnabbaxdxtan2106．abcosx＝

tanabbaCabba(a2b2)

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dx1barctan(tanx)C2222acosxbsinxaba107．＝ 1btanxadxlnC22222abbtanxa108．acosxbsinx＝

11sinaxxcosaxCxsinaxdx2aa109．＝

110．

2xsinaxdx1222xcosax2xsinax3cosaxCaa＝a 11cosaxxsinaxCxcosaxdx2a111．＝a

1222xsinaxxcosaxsinaxCxcosaxdx23aaa112．＝

2（十二）含有反三角函数的积分（其中a0）

xxarcsindxxarcsina2x2Ca＝a113．

x2a2xx2x2()arcsinaxCxarcsindx24a4a114．＝ x3x12xarcsin(x2a2)a2x2Cxarcsindxa9a＝3115．

2xx22arccosdxxarccosaxCa＝a116．

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x2a2xx2x2()arccosaxCxarccosdx4a4a＝2117． x3x12x222arccos(x2a)axCxarccosdxa9a＝3118．

2xxaarctandxxarctanln(a2x2)Ca＝a2119．

x12xa2xarctandx(ax)arctanxCa＝2a2120．

x3xa2a3x22arctanxln(ax)Cxarctandxa66a＝3121．

2（十三）含有指数函数的积分

1xaCadxlna122．＝

x1axeCedxa123．＝

ax1(ax1)eaxCxedx2124．＝a

ax125．

naxxedx1naxnn1axxexedxa＝a

xx1xaaC2xadxlna(lna)126．＝

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1nxnn1xxaxadxxadxlnalna127．＝

nx1eax(asinbxbcosbx)Cesinbxdx22128．＝ab

ax1eax(bsinbxacosbx)Cecosbxdx22129．＝ab

ax130．

axnesinbxdx1axn1esinbx(asinbxnbcosbx)222＝abn

n(n1)b2axn22esinbxdx22abn

1eaxcosn1bx(acosbxnbsinbx)ecosbxdx222131．＝abn

axnn(n1)b2ax2ecosn2bxdx22abn

（十四）含有对数函数的积分 132．

dxlnlnxC133．xlnx＝

lnxdx＝xlnxxC

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1n11x(lnx)Cxlnxdxn1n1134．＝

nn(lnx)dx135．＝

x(lnx)nn(lnx)dxn1

1nm1nx(lnx)xm(lnx)n1dxx(lnx)dxmn136．＝m1（十五）含有双曲函数的积分 137．shxdx＝chxC

138．chxdx＝shxC

139．thxdx＝lnchxC

sh2140．xdxx＝214sh2xC

ch2141．xdxx＝214sh2xC

（十六）定积分 142．cosnxdx＝sinnxdx＝0

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m1

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143．cosmxsinnxdx＝0

0,mncosmxcosnxdx,mn144．＝



0,mnsinmxsinnxdx,mn145．＝



0,mn,mnsinmxsinnxdxcosmxcosnxdx146．0＝0＝2

2020147．

In＝sinxdxn＝cosnxdx

n1In2In＝n

n1n342Lnn253（n为大于1的正奇数），I1＝1

InInn1n331Lnn2422（n为正偶数），I0＝2

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