积分公式大全

（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．

x1＝dxaxba2(axbblnaxb)C

x211dx＝3(axb)22b(axb)b2lnaxbC 4．a2axb5．1axbdx＝x(axb)blnxC 6．

1aaxbdxlnC ＝22bxbxx(axb)7．

x1bdx＝(lnaxb)C (axb)2a2axbx21b2dx＝3(axb2blnaxb)C 8．2(axb)aaxb9．

11axbdxlnC ＝

xx(axb)2b(axb)b2（二）含有axb的积分

23(axb)C 3a2311．xaxbdx＝(3ax2b)(axb)C 215a22(15a2x212abx8b2)(axb)3C 12．xaxbdx＝3105a10．

axbdx＝13．

x2dx＝2(ax2b)axbC

3aaxb文档

14．

x22dx＝(3a2x24abx8b2)axbC 315aaxbdx15．＝xaxb16．

1lnbaxbbC(b0)axbb

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

axbdxdx＝2axbb xxaxbaxbaxbadxdx＝ x2x2xaxb2218．

（三）含有xa的积分 19．

dx1x=arctanC x2a2aadxx2n3dx=(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n1 1xadxln=x2a22axaC

220．

21．

（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bx12dx＝ax2b2alnaxbC

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24．x2ax2bdx＝xabadxax2b

．dxx(ax2b)＝1x2252blnax2bC

26．

dx1x2(ax2b)＝bxadxbax2b 27．dxaax2b1x3(ax2b)＝2b2lnx22bx2C 28．

dx(ax2b)2＝x2b(ax2b)1dx2bax2b

（五）含有ax2bxc(a0)的积分

22ax2arctanb29．dx4acb4acb2Cax2bxc＝12axbb24b24aclnac2axbb24acC30．

xax2bxcdx＝12bdx2alnaxbxc2aax2bxc（六）含有x2a2(a0)的积分 31．

dx＝x2a2arshxaC1＝ln(xx2a2)C 32．

dxx(x2a2)3＝a2x2a2C

33．

xx2a2dx＝x2a2C

34．

x1(x2a2)3dx＝x2a2C

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(b24ac)(b24ac)

35．

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

x2(x2a2)3dxdx＝xx2a2ln(xx2a2)C

1x2a2aC 37．＝ln22axxxa38．

x2x2a2C ＝222axxadx39．

x2a22xaln(xx2a2)C xadx＝2222x342232222(xa)dx＝(2x5a)xaaln(xx2a2)C 8812241．xxadx＝(x2a2)3C

340．42．x2xa42222xadx＝(2xa)xaln(xx2a2)C

882243．

x2a2x2a2a22dx＝xaalnC xxx2a2x2a2dx＝ln(xx2a2)C 2xx44．

（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

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48．

x(xa)223dx＝1xa22C

49．

xxx2a22xalnxx2a2C dx＝22x2a2x250．

x2(x2a2)3dxx2a2dxdx＝xx2a2lnxx2a2C

51．＝1aarccosC ax52．

2x2a2C ＝222axxa2253．

x2a22xalnxx2a2C xadx＝22x34223222222(xa)dx＝(2x5a)xaalnxxaC 8812255．xxadx＝(x2a2)3C

354．56．x2xa42222xadx＝(2xa)xalnxx2a2C

882257．

ax2a2dx＝x2a2aarccosC

xxx2a2x2a222dxlnxxaC ＝2xx58．

（八）含有a2x2(a0)的积分 59．

dxa2x2＝arcsinxC axC

60．

dx(ax)223＝a2ax22文档

61．

xa2x2dx＝a2x2C

1ax2262．

x(ax)223dx＝C

63．

x2a2x2axarcsinC dx＝22aa2x2x264．

x2(a2x2)3dxdx＝xa2x2arcsinxC a1aa2x2C 65．＝ln22axxax66．

x2a2x2C ＝222axaxdx67．

x2a2x2axarcsinC axdx＝22a22x34x2232222(ax)dx＝(5a2x)axaarcsinC 88a12269．xaxdx＝(a2x2)3C

368．70．x2xa4x2222axdx＝(2xa)axarcsinC

88a2271．

a2x2aa2x222dx＝axalnC xxa2x2a2x2xdxarcsinC ＝x2xa72．

（九）含有ax2bxc(a0)的积分 73．

dxax2bxc＝1ln2axb2aax2bxcC a文档

74．

ax2bxcdx＝2axbax2bxc 4a

xax2bxc4acb28a3ln2axb2aax2bxcC

75．

dx＝1ax2bxc ab2a3ln2axb2aax2bxcC

dxcbxax2276．

＝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a(ba)ln(xadx＝(xb)xbxb80．

xaxaxa(ba)arcsinC dx＝(xb)bxbxbxxadxC＝2arcsinbx(xa)(bx)81．

(ab)

82．

2xab(ba)2xa(xa)(bx)arcsinC (xa)(bx)dx＝44bx (ab) （十一）含有三角函数的积分 83．sinxdx＝cosxC

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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1nn295．sinxdx＝sinxcosxsinxdx nn1n1nn1n296．cosxdx＝cosxsinxcosxdx nndx1cosxn2dx97．＝ sinnxn1sinn1xn1sinn2xdx1sinxn2dx98．＝ nn1n2cosxn1cosxn1cosx1m1mnm2n99．cosxsinxdx＝cosm1xsinn1xcosxsinxdx mnmn1n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)文档

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)103．

2dx＝absinxa2b2arctanatanxb2C22ab(a2b2)

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

(a2b2)

2ababxdxarctan(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)

dx1b＝arctan(tanx)C a2cos2xb2sin2xaba1btanxadxln＝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＝

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x2a2xx2xax2C 114．xarcsindx＝()arcsin24a4ax3x12x222115．xarcsindx＝arcsin(x2a)axC

3a9a2116．arccosdx＝xarccosxaxa2x2C ax2a2xx2xax2C 117．xarccosdx＝()arccos24a4ax3x12x222118．xarccosdx＝arccos(x2a)axC

3a9a2xxa＝dxxarctanln(a2x2)C aa2x12xa2120．xarctandx＝(ax)arctanxC

a2a2119．arctanx3xa2a3xln(a2x2)C 121．xarctandx＝arctanx3a66a2（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 lnalna1axeax(asinbxbcosbx)C 128．esinbxdx＝22ab1axaxe(bsinbxacosbx)C 129．ecosbxdx＝22ab127．xadx＝

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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11n134．xlnxdx＝x(lnx)C

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文档

0,mn145．sinmxsinnxdx＝

,mn146．

0sinmxsinnxdx＝2000,mncosmxcosnxdx＝

,mn2147． In＝ In＝

sinxdx＝cosnxdx

n20n1In2 nn1n3 Innn2n1n3Innn2

42，I1＝1  （n为大于1的正奇数）

5331，I0＝ （n为正偶数）

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