高等数学积分公式大全(2)

61．

??xa2?x2xdx＝?a2?x2?C

1a?x2262．

(a?x)223dx＝?C

63．

??x2a2x2dx＝?a?x?arcsin?C

22aa2?x2x2(a2?x2)3dx＝x264．

xa2?x2?arcsinx?C a1a?a2?x2?C 65．?＝ln22axxa?xdx66．

?x2a2?x2?C ＝?222axa?xdx2267．

?x2a2x2a?x?arcsin?C a?xdx＝22ax34x2222223＝(5a?2x)a?x?aarcsin?C (a?x)dx?88a169．?xa2?x2dx＝?(a2?x2)3?C

368．70．x?2xa4x2222a?xdx＝(2x?a)a?x?arcsin?C

88a2271．

?a?a2?x2a2?x222?C dx＝a?x?alnxx72．

?a2?x2a2?x2xdx??arcsin?C ＝x2xa2（九）含有?ax?bx?c(a?0)的积分 73．

?dxax2?bx?c＝1ln2ax?b?2aax2?bx?c?C a74．

?ax2?bx?cdx＝2ax?bax2?bx?c 4a?24ac?b

8a3ln2ax?b?2a2ax?bx?c? C75．

?xax2?bx?cdx＝1ax2?bx?c ab2a3

?dxc?bx?ax22ln2ax?b?2a2ax?b?xc? C76．

?＝?12ax?barcsin?C

2ab?4ac77．

??2ax?bb2?4ac2ax?b2c?bx?ax?arcsin?C c?bx?axdx＝324a8ab?4ac78．

xc?bx?ax2dx＝?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?xx?adx?C＝2arcsinb?x(x?a)(b?x)81．

(a?b)

82．

2x?a?b(b?a)2x?a(x?a)(b?x)dx＝(x?a)(b?x)?arcsin?C

44b?x (a?b) （十一）含有三角函数的积分 83．sinxdx＝?cosx?C

?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

????22x1?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?1＝?cosm?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)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?1a?bdx2ln106．?＝a?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＝

x2a2xx2x114．?xarcsindx＝(?)arcsin?a?x2?C

24a4ax3x12x222115．?xarcsindx＝arcsin?(x?2a)a?x?C

3a9a2116．arccosdx＝xarccos?xax?a2?x2?C ax2a2xx2x117．?xarccosdx＝(?)arccos?a?x2?C

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?1ax128．?esinbxdx＝2eax(asinbx?bcosbx)?C 2a?b1axax129．?ecosbxdx＝2e(bsinbx?acosbx)?C 2a?b127．xadx＝

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