高等数学积分公式大全

（一）含有axb的积分(a1．

0)

dx1＝axbalnaxbC

(axb)dx＝

2．

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．(axb)2aaxb9．

11axbdxlnC ＝22b(axb)bxx(axb)（二）含有axb的积分

2(axb)3C 3a2311．xaxbdx＝(3ax2b)(axb)C 215a2212．xaxbdx＝(15a2x212abx8b2)(axb)3C 3105a10．

axbdx＝13．

x2dx＝2(ax2b)axbC

3aaxbx22222dx＝(3ax4abx8b)axbC 315aaxb14．

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＝ 2xx2xaxb218．

（三）含有x19．

a2的积分

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bx12＝dxlnaxbC ax2b2ax2xbdxdx＝224．2

axbaaaxbdx1x2lnC 25．＝

x(ax2b)2bax2b26．

dx1adx＝x2(ax2b)bxb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dx4acb2arctanb4acb2C29．ax2bxc＝12axbb24acb24acln2axbb24acC30．

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

31．

dx＝arshxx2a2aC1＝ln(xx2a2)C 32．

dxx(x2a2)3＝a2x2a2C

33．

xx2a2dx＝x2a2C

34．

x(x2a2)3dx＝1x2a2C

35．

x2xx2a2dx＝2x2a2a22ln(xx2a2)C 236．

x(x2a2)3dx＝xx2a2ln(xx2a2)C

．dx1x2a237axx2a2＝alnxC 38．

dx2＝x2a2x2ax2a2xC 39．

x2a2dx＝xx2aa22ln(xx2a222)C (b24ac)(b24ac)

x34223222222(xa)dx＝(2x5a)xaaln(xxa)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a222dxln(xxa)C ＝2xx44．

（七）含有x2a2(a0)的积分

＝

45．

dxx2a2xxarchC1=lnxx2a2C xa＝46．

dx(xa)xx2a2223xa2xa22C

47．

dx＝x2a2C

48．

x(xa)223dx＝1xa22C

49．

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

x2(x2a2)3dxx2a2dx＝xx2a2lnxx2a2C

51．＝1aarccosC ax52．

x2x2a2C ＝222axxadx53．

x2a22xalnxx2a2C xadx＝222254．

x3(x2a2)3dx＝(2x25a2)x2a2a4lnxx2a2C

8855．xxadx＝221(x2a2)3C 356．x2xa42222x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

＝arcsin59．

dxa2x2xC axC

60．

dx(ax)xa2x2223＝a2ax2261．

dx＝a2x2C

62．

x(ax)223dx＝1ax22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 ＝88a122(a2x2)3C 69．xaxdx＝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

dx＝73．

ax2bxc1ln2axb2aax2bxcC a74．

ax2bxcdx＝xax2bxcdxcbxax222axbax2bxc 4a1ax2bxc a12axbarcsinC

2ab4ac75．

dx＝76．

＝77．

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

324a8ab4acxcbxax278．

dx＝1b2axbcbxax2arcsinC

32a2ab4ac（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x81．

xadxC＝2arcsinbx(xa)(bx)(ab)

82．

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

83．84．85．86．

sinxdx＝cosxC cosxdx＝sinxC tanxdx＝lncosxC cotxdx＝lnsinxC

xlntan()C＝lnsecxtanxC ＝secxdx4287．

88．

cscxdx＝lntansec2xC＝lncscxcotxC 289．90．91．92．93．

xdx＝tanxC

2cscxdx＝cotxC

secxtanxdx＝secxC cscxcotxdx＝cscxC

2sinxdx＝

x1sin2xC 24x1294．cosxdx＝sin2xC

241n1n1nn295．sinxdx＝sinxcosxsinxdx nn1n1nn1n296．cosxdx＝cosxsinxcosxdx

nndx1cosxn2dx97．＝ sinnxn1sinn1xn1sinn2xdx1sinxn2dx98．＝ nn1n2cosxn1cosxn1cosx1m1mnm2n99．cosxsinxdx＝cosm1xsinn1xcosxsinxdx

mnmn1n1cosm1xsinn1xcosmxsinn2xdx ＝mnmn100．

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

2(ab)2(ab)11sin(ab)xsin(ab)xC

2(ab)2(ab)101．

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bsinx22baatanbba2atan105．

(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 2aa12222110．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＝

x2a2xx2x)arcsinax2C 114．xarcsindx＝(24a4ax3x1xarcsin(x22a2)a2x2C 115．xarcsindx＝3a9a2116．

xx22arccosdxxarccosaxC ＝aax2a2xx2x)arccosax2C 117．xarccosdx＝(24a4ax3x1xarccos(x22a2)a2x2C 118．xarccosdx＝3a9a2xxa22＝arctandxxarctanln(ax)C aa2x12xa2120．xarctandx＝(ax)arctanxC

a2a2119．

x3xa2a3xarctanxln(a2x2)C 121．xarctandx＝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．

xadx＝

126．

xxadx＝

xx1xaaC 2lna(lna)1nxnn1xxaxadx lnalna1axax128．esinbxdx＝2e(asinbxbcosbx)C 2ab1ax129．ecosbxdx＝2eax(bsinbxacosbx)C 2ab1axnaxn1130．esinbxdx＝2esinbx(asinbxnbcosbx) 22abn1axnaxn1131．ecosbxdx＝2ecosbx(acosbxnbsinbx) 22abn127．

nxxadx＝

（十四）含有对数函数的积分 132．133．

lnxdx＝xlnxxC

dxxlnx＝lnlnxC

1n11nx(lnx)C 134．xlnxdx＝

n1n1135．

nnx(lnx)n(lnx)dx (lnx)dx＝n1136．

mnx(lnx)dx＝1nmn1xm1(lnx)nx(lnx)dx m1m1（十五）含有双曲函数的积分 137．138．139．140．

shxdx＝chxC chxdx＝shxC thxdx＝lnchxC

2shxdx＝x1sh2xC 24x12141．chxdx＝sh2xC

24（十六）定积分 142．143．

cosnxdx＝sinnxdx＝0

cosmxsinnxdx＝0

144．

0,mncosmxcosnxdx＝ ,mn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Lnn2n1n3InLnn242 （n为大于1的正奇数），I1＝1 5331（n为正偶数），I0＝ 4222