积分公式大全

（一）含有axb的积分(a0)

dx1lnaxbC1．axb＝a

1(axb)1C(axb)dx2．＝a(1)（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＝ 1axbdxlnCx(axb)bx5．＝ 1aaxbdxlnC2x2(axb)x6．＝bxb

x1bdx(lnaxb)C2(axb)2aaxb7．＝

1

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

11axbdxlnCx(axb)2b(axb)b2x9．＝

（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

14．

x22222dx(3ax4abx8b)axbCaxb＝15a3

dxxaxb15．＝

1lnbaxbbC(b0)axbb2axbarctanC(b0)bb

2

216．xdxax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

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2axa21．xa=

2（四）含有axb(a0)的积分

1arctanab1lndx2ab2axb22．＝

axCb(b0)axbC(b0)axb 3

x23．ax2bdx1＝2alnax2bC

x224．ax2bdxx＝abdxaax2b

dx1lnx225．x(ax2b)＝2bax2bC

26．dx1adxx2(ax2b)＝bxbax2b

27．dxaax2b1x3(ax2b)＝2b2lnx22bx2C

28．dxx1d(ax2b)2＝2b(ax2b)x2bax2b

（五）含有ax2bxc(a0)的积分

22axb4acb2arctan4acb2Cdx1ln2axbb24ac29．ax2bxc＝b24ac2axbb24acC4

(b24ac)(b24ac)

x1bdx2dxlnaxbxc22aax2bxc 30．axbxc＝2a22（六）含有xa(a0)的积分

31．

dxx2a2＝

arshxC122ln(xxa)C a＝

32．

dx(x2a2)3x＝a2xa22C

33．

xxa22dx22＝xaC

34．

x(x2a2)3dx＝

1xa22C

35．

2xa22dxxaln(xx2a2)C22xa2＝2

x236．

x2(xa)223dx＝

xx2a2ln(xx2a2)C

37．

xdxx2a2＝1x2a2alnCax

5

38．

x2x2a2C222xa＝ax

dx39．2xa22xaln(xx2a2)Cx2a2dx2＝2

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

22341．

22xxadx1(x2a2)3C＝3

42．

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

43．

x2a2a22x2a2xaalnCdxxx＝

44．

x2a2x2a2dxln(xx2a2)C2xx＝

22（七）含有xa(a0)的积分

45．

dxxa＝

22xxarchC1xa=

lnxx2a2C

6

46．

dx(x2a2)3＝ax2xa22C

47．

xxa22dx22xaC ＝48．

x(xa)223dx＝

1xa22C

49．

2xa22dxxalnxx2a2C22xa2＝2

x250．

x2(x2a2)3dx＝

xx2a2lnxx2a2C

51．

xdxx2a2＝1aarccosCax

52．

x2x2a2Cx2a2＝a2x

dx53．2xa22xalnxx2a2Cx2a2dx2＝2

．

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

2237

1(x2a2)3Cxxadx55．＝3

22x56．24xa2222(2xa)xalnxx2a2Cx2a2dx8＝8

57．

a22x2a2xaaarccosCdxxx＝

58．

x2a2x2a2dxlnxx2a2C2xx＝

22（八）含有ax(a0)的积分

59．

dxa2x2＝

arcsinxCa

60．

dx(a2x2)3x＝a2ax22C

61．

xax22dx22＝axC

62．

x(a2x2)3dx1＝ax22C

8

63．

2xax22dxaxarcsinCa2x222a＝

x2．

x2(a2x2)3dxx22＝axarcsinxCa

65．

xdxa2x2＝1aa2x2lnCax

66．

x2a2x2C222ax＝ax

dx67．2xax2222axarcsinCaxdx2a＝2

68．(a2x2)3dxx3x(5a22x2)a2x2a4arcsinC8a＝8

69．

22xaxdx1(a2x2)3C＝3 70．

2x4xax222222(2xa)axarcsinCaxdx8a＝8

71．

aa2x222a2x2axalnCdxxx＝

9

72．

a2x2a2x2xdxarcsinC2xxa＝

2axbxc(a0)的积分 （九）含有73．

12ln2axb2aaxbxcC2aaxbxc＝

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

75．

xaxbxc2dx1ax2bxc＝a

b2a3ln2axb2aax2bxcC

76．

dxcbxax2＝

12axbarcsinC2ab4ac

77．2axbb24ac2axb2cbxaxarcsinC2cbxaxdx328ab4ac＝4a

10

78．

xcbxax2dx1＝acbxax2b2axb2a3arcsinb24acC xa（十）含有xb或(xa)(bx)的积分

xa79．

xbdx＝(xb)xaxb(ba)ln(xaxb)C

a80．

xbxdxxaxa＝(xb)bx(ba)arcsinbxC

81．

dx(xa)(bx)＝

2arcsinxabxC(ab)

(xa)(bx)dxxab82．2＝4(xa)(bx)(ba)2xa4arcsinbxC （十一）含有三角函数的积分

83．sinxdx＝cosxC

84．cosxdx＝sinxC

11

(ab)

tanxdxlncosxC85．＝

86．cotxdx＝lnsinxC

xlntan()Csecxdx4287．＝＝lnsecxtanxC

cscxdx88．＝

lntanxC2＝lncscxcotxC

sec．2xdx＝tanxC

90．

2cscxdx＝cotxC

91．secxtanxdx＝secxC

92．cscxcotxdx＝cscxC

x1sin2xCsinxdx2493．＝

2x1sin2xCcosxdx2494．＝

2 12

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

n1n1n1n2cosxsinxcosxdxcosxdxnn96．＝

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

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

1m1m1n1m2ncosxsinxcosxsinxdxcosxsinxdxmn99．＝mn

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

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

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

13

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

x22bba12lnC22dxxbaatanbb2a22104．absinx＝atan(a2b2)

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

x1ab2lnabbaxdxtan2106．abcosx＝

tanabbaCabba(a2b2)

dx1barctan(tanx)C2222a107．acosxbsinx＝ab

1btanxadxlnC22222abbtanxa108．acosxbsinx＝

11sinaxxcosaxCxsinaxdx2aa109．＝

1222xcosaxxsinaxcosaxCxsinaxdx23aaa110．＝

2 14

11cosaxxsinaxCxcosaxdx2aa111．＝

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．

x2a2xx2x()arcsinax2Cxarcsindx4a4a＝2114．

x3x12x222arcsin(x2a)axCxarcsindxa9a＝3115．

2xx22arccosdxxarccosaxCa＝a116．

x2a2xx2x2()arccosaxCxarccosdx4a4a＝2117．

x3x12x222arccos(x2a)axCxarccosdxa9a＝3118．

2 15

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

x12xa2xarctandx(ax)arctanxCa2a2120．＝

x3xa2a3xarctanxln(a2x2)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

ax1naxnn1axxexedxxedxaa125．＝

naxxx1xaaC2xadxlna(lna)126．＝

x 16

1nxnxaxn1axdxxadxlna127．＝lna

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

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

ax1axn1esinbx(asinbxnbcosbx)esinbxdx222130．＝abn

axnn(n1)b2axn22esinbxdx22abn

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

axnn(n1)b2axn22ecosbxdx22 abn

（十四）含有对数函数的积分

lnxdx132．＝xlnxxC

dxlnlnxC133．xlnx＝

17

1n11x(lnx)Cxlnxdxn1n1134．＝

n135．

n(lnx)dx＝

x(lnx)nn(lnx)dxn1

1nm1nmn1x(lnx)x(lnx)dxx(lnx)dxm1136．＝m1

mn（十五）含有双曲函数的积分

137．shxdx＝chxC

138．chxdx＝shxC

thxdx139．＝lnchxC

x1sh2xCshxdx140．＝24

2141．

2chxdxx1sh2xC＝24

（十六）定积分

18

142．cosnxdx＝sinnxdx＝0

143．cosmxsinnxdx＝0

0,mncosmxcosnxdx,mn144．＝

0,mnsinmxsinnxdx,mn145．＝

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

147．

In＝20sinxdxn＝20cosnxdx

n1In2In ＝n

Inn1n3nn24253 （n为大于1的正奇数），I1＝1

Inn1n3nn231422（n为正偶数），I0＝2

19