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积分大全

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常 用 积 分 公 式
（一）含有axb的积分(a0 1．dx1＝axbalnaxbC

2．(axbdx＝1(axb1C（1）
a(13．x1＝dxaxba2(axbblnaxbC
x211dx＝3(axb22b(axbb2lnaxbC 4．a2axb5．1axbdx＝x(axbblnxC
6．1aaxbdxlnC ＝22bxbxx(axb7．x1bdx＝(lnaxbC (axb2a2axbx21b2dx＝3(axb2blnaxbC 8．2(axbaaxb9．11axbdxlnC ＝xx(axb2b(axbb2（二）含有axb的积分
23(axbC 3a2311．xaxbdx＝(3ax2b(axbC
215a22(15a2x212abx8b2(axb3C 12．xaxbdx＝3105a10．axbdx＝13．x2dx＝2(ax2baxbC
3aaxb
14．x22dx＝(3a2x24abx8b2axbC
315aaxbdx15．＝xaxb16．1lnbaxbbC(b0axbb
2axbarctanC(b0bbx2dxaxbadx＝ bx2bxaxbaxb17．axbdxdx＝2axbb xxaxbaxbaxbadxdx＝ x2x2xaxb2218．（三）含有xa的积分 19．dx1x=arctanC x2a2aadxx2n3dx=(x2a2n2(n1a2(x2a2n12(n1a2(x2a2n1 1xadxln=x2a22axaC
220．21．（四）含有axb(a0的积分
1arctandxab22．2＝axb1ln2ab23．axCb(b0
axbC(b0axbx12dx＝ax2b2alnaxbC

24．x2ax2bdx＝xabadxax2b
．dxx(ax2b＝1x2252blnax2bC
26．dx1ax2(ax2b＝bxdxbax2b
27．dxaax2b1x3(ax2b＝2b2lnx22bx2C 28．dxx(ax2b2＝2b(ax2b1dx2bax2b
（五）含有ax2bxc(a0的积分
22ax2arctanb29．dx4acb4acb2Cax2bxc＝12axbb2ln4acCb24ac2axbb24ac30．x1bdxax2bxcdx＝2alnax2bxc2aax2bxc（六）含有x2a2(a0的积分 31．dxx2a2＝arshxaC1＝ln(xx2a2C 32．dxx(x2a23＝a2x2a2C
33．xx2a2dx＝x2a2C
34．x1(x2a23dx＝x2a2C
(b24ac(b24ac

35．x2a22xaln(xx2a2C dx＝22x2a2x236．x2(x2a23dxdx＝xx2a2ln(xx2a2C
1x2a2aC 37．＝ln22axxxa38．x2x2a2C ＝222axxadx39．x2a22xaln(xx2a2C xadx＝2222x342232222(xadx＝(2x5axaaln(xx2a2C 8812241．xxadx＝(x2a23C
340．42．x2xa42222xadx＝(2xaxaln(xx2a2C
882243．x2a2x2a2a22dx＝xaalnC xxx2a2x2a2dx＝ln(xx2a2C 2xx44．（七）含有x2a2(a0的积分 45．dxx2a2＝xxarchC1=lnxx2a2C xa＝46．dx(xaxx2a2223xa2xa22C
47．dx＝x2a2C

48．x(xa223dx＝1xa22C
49．xxx2a22xalnxx2a2C dx＝22x2a2x250．x2(x2a23dxx2a2dxdx＝xx2a2lnxx2a2C
51．＝1aarccosC ax52．2x2a2C ＝222axxa2253．x2a22xalnxx2a2C xadx＝22x34223222222(xadx＝(2x5axaalnxxaC 8812255．xxadx＝(x2a23C
354．56．x2xa42222xadx＝(2xaxalnxx2a2C
882257．ax2a2dx＝x2a2aarccosC
xxx2a2x2a222dxlnxxaC ＝2xx58．（八）含有a2x2(a0的积分 59．dxa2x2＝arcsinxC
axC
60．dx(ax223＝a2ax22
61．xa2x2dx＝a2x2C
1ax2262．x(ax223dx＝C
63．x2a2x2axarcsinC dx＝22aa2x2x264．x2(a2x23dxdx＝xa2x2arcsinxC
a1aa2x2C 65．＝ln22axxax66．x2a2x2C ＝222axaxdx67．x2a2x2axarcsinC axdx＝22a22x34x2232222(axdx＝(5a2xaxaarcsinC 88a12269．xaxdx＝(a2x23C
368．70．x2xa4x2222axdx＝(2xaaxarcsinC
88a2271．a2x2aa2x222dx＝axalnC xxa2x2a2x2xdxarcsinC ＝x2xa72．（九）含有ax2bxc(a0的积分 73．dxax2bxc＝1ln2axb2aax2bxcC
a
74．ax2bxcdx＝2axbax2bxc
4a
xax2bxc4acb28a3ln2axb2aax2bxcC
75．dx＝1ax2bxc
ab2a3ln2axb2aax2bxcC

dxcbxax2276．＝12axbarcsinC
2ab4ac77．2axbb24ac2axb2cbxaxdx＝cbxaxarcsinC
324a8ab4acxcbxax2dx＝1b2axbcbxax2arcsinC
32a2ab4ac78．（十）含有xa或(xa(bx的积分 xbxbC
79．xaxadx＝(xb(baln(xaxbxb80．xaxaxadx＝(xb(baarcsinC bxbxbxxadxC＝2arcsinbx(xa(bx81．(ab
82．2xab(ba2xa(xa(bxarcsinC (xa(bxdx＝44bx (ab （十一）含有三角函数的积分 83．sinxdx＝cosxC

84．cosxdx＝sinxC 85．tanxdx＝lncosxC 86．cotxdx＝lnsinxC 87．secxdx＝lntan(xC＝lnsecxtanxC 4288．cscxdx＝lntanxC＝lncscxcotxC
289．secxdx＝tanxC 90．cscxdx＝cotxC 91．secxtanxdx＝secxC 92．cscxcotxdx＝cscxC
22x1sin2xC 24x1294．cosxdx＝sin2xC
241n1n1nn295．sinxdx＝sinxcosxsinxdx nn1n1nn1n296．cosxdx＝cosxsinxcosxdx nndx1cosxn2dx97．＝ sinnxn1sinn1xn1sinn2xdx1sinxn2dx98．＝ nn1n2cosxn1cosxn1cosx1m1mnm2n99．cosxsinxdx＝cosm1xsinn1xcosxsinxdx mnmn1n1cosm1xsinn1xcosmxsinn2xdx ＝mnmn93．sinxdx＝2100．sinaxcosbxdx＝11cos(abxcos(abxC
2(ab2(ab
101．sinaxsinbxdx＝11sin(abxsin(abxC
2(ab2(ab102．cosaxcosbxdx＝11sin(abxsin(abxC
2(ab2(ab103．2dx＝absinxa2b2arctanatanxb2C22ab(a2b2
x22bbadx12104．＝lnC22xabsinxbaatanbb2a22atan105．(a2b2
2ababxdxarctan(tanC＝abcosxababab2(a2b2
xdx1ab2106．＝lnabcosxabbaxtan2tan107．abbaCabba(a2b2
dx1b＝arctan(tanxC a2cos2xb2sin2xaba1btanxadxln＝a2cos2xb2sin2x2abbtanxaC
108．11sinaxxcosaxC a2a12222110．xsinaxdx＝xcosax2xsinax3cosaxC
aaa11111．xcosaxdx＝2cosaxxsinaxC
aa12222112．xcosaxdx＝xsinax2xcosax3sinaxC
aaa（十二）含有反三角函数的积分（其中a0）
xx22113．arcsindx＝xarcsinaxC
aa109．xsinaxdx＝
x2a2xx2xax2C 114．xarcsindx＝(arcsin24a4ax3x12x222115．xarcsindx＝arcsin(x2aaxC
3a9a2116．arccosdx＝xarccosxaxa2x2C
ax2a2xx2xax2C 117．xarccosdx＝(arccos24a4ax3x12x222118．xarccosdx＝arccos(x2aaxC
3a9a2xxa＝dxxarctanln(a2x2C aa2x12xa2120．xarctandx＝(axarctanxC
a2a2119．arctanx3xa2a3xln(a2x2C 121．xarctandx＝arctanx3a66a2（十三）含有指数函数的积分
1xaC lna1axax123．edx＝eC
a1axax124．xedx＝2(ax1eC
a1naxnn1axnax125．xedx＝xexedx
aa122．adx＝x126．xadx＝nxxxx1aaxC 2lna(lna1nxnn1xxaxadx lnalna1axeax(asinbxbcosbxC 128．esinbxdx＝22ab1axaxe(bsinbxacosbxC 129．ecosbxdx＝22ab127．xadx＝
130．esinbxdx＝axn1eaxsinn1bx(asinbxnbcosbx 222abnn(n1b2axn2esinbxdx 2ab2n2131．ecosbxdx＝axn1axn1ecosbx(acosbxnbsinbx 222abnn(n1b2axn2ecosbxdx 222abn（十四）含有对数函数的积分 132．lnxdx＝xlnxxC
dxxlnx＝lnlnxC
1n11n134．xlnxdx＝x(lnxC
n1n1133．135．(lnxdx＝x(lnxn(lnx136．x(lnxdx＝nnn1dx
mn1nmn1xm1(lnxnx(lnxdx m1m1（十五）含有双曲函数的积分 137．shxdx＝chxC 138．chxdx＝shxC 139．thxdx＝lnchxC
x1sh2xC 24x12141．chxdx＝sh2xC
24140．shxdx＝2（十六）定积分 142．143．cosnxdx＝sinnxdx＝0 cosmxsinnxdx＝0 144．0,mncosmxcosnxdx＝ ,mn
0,mn145．sinmxsinnxdx＝
,mn146．0sinmxsinnxdx＝2000,mncosmxcosnxdx＝
,mn2147． In＝ In＝sinxdx＝cosnxdx
n20n1In2
nn1n342 In，I1＝1 L （n为大于1的正奇数）nn253n1n331，I0＝ InL（n为正偶数）nn24222


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