高等數(shù)學(xué)積分公式大全

常用積分公式（一）含有axb的積分(a0)1．dx＝1lnaxbCaxba2．(axb)dx＝1(axb)1C（1）a(1)3．xd＝1(axbblnaxb)C2axbxa4．x2dx＝11(axb)22b(axb)b2lnaxbCaxba325．dxb)＝1lnaxbCx(axbx6．2dx＝1a2lnaxbCx(axb)bxbxx2dx＝1b)C7．(axb)a2(lnaxbaxb8．(axx22dx＝13(axb2blnaxbb2)Cb)aaxbdx2＝11axbC9．x(axb)b(axb)b2lnx（二）含有axb的積分10．a(chǎn)xbdx＝2(axb)3C3a11．xaxbdx＝2(axb)3C15a2(3ax2b)12．x2axbdx＝2(15a2x212abx8b2)(axb)3C105a313．xbd＝2(ax2b)axbCaxx3a214．x2bdx＝23(3a2x24abx8b2)axbCax15adx1lnaxbbC(b0)15．＝baxbbaxbx2arctanaxbC(b0)bb16．dx＝axbadxbx2bxaxbx2axb17．a(chǎn)xbd＝2axbbdxxaxbx18．a(chǎn)xbdx＝axbadxbx2x2xax（三）含有x2a2的積分19．dx=1arctanxCx2a2aadxx2n3dx20．(x2a2)n=2(n1)a2(x2a2)n12(n1)a2(x2a2)n121．dxa2=1lnxaCx22axa（四）含有ax2b(a0)的積分1arctanaC(b0)dxxabb22．2＝axbaxb1ln(b0)2abaxCb23．xdx＝1lnax2bCax2b2a24．x2xbdx2dx＝aax2baxba25．dx＝1lnx2Cx(ax2b)2bax2b26．dx＝1adxbxbax2bx2(ax2b)dxaax2b1C27．x3(ax2＝2lnx22b)2b2bx28．dxx1dx(ax2b)2＝2b(ax2b)2bax2b（五）含有ax2bxc(a0)的積分2arctan2axbC(b24ac)29．dx＝4acb24acb2ax2b2bxcb21ln2axb4acC(b24ac)4ac2axbb24ac30．a(chǎn)x2xdx＝1lnax2bxcbdxcbxc2a2aax2bx（六）含有x2a2(a0)的積分31．dxa2＝arshxC1＝ln(xx2a2)Cx2a32．dx＝xC(x2a2x2a2a2)333．x2xa2dx＝x2a2C34．xdx＝1C(x2x2a2a2)335．x2a2dx＝xx2a2a2ln(xx2a2)Cx22236．x2dx＝x2xa2ln(xx2a2)C(x2a2)337．dxa2＝1lnx2a2aCxx2ax38．dx＝x2a2Cx2x2a2xa239．x2a2dx＝xx2a2a2ln(xx2a2)C2240．(x2a2)3dx＝x(2x25a2)x2a23a4ln(xx2a2)C8841．xx2a2dx＝1(x2a2)3C342．x2x2a2dx＝x(2x2a2)x2a2a4ln(xx2a2)C8843．x2xa2dx＝x2a2alnx2a2aCx44．x2a2dx＝x2a2ln(xx2a2)Cx2x（七）含有x2a2(a0)的積分45．dx＝xarchxC1=lnxx2a2Cx2a2xa46．dx＝xC(x2a2x2a2a2)347．x2xa2dx＝x2a2C48．xdx＝1C(x2x2a2a2)349．x2a2dx＝xx2a2a2lnxx2a2Cx22250．x2dx＝xa2lnxx2a2C(x2a2)3x251．dx＝1arccosaCxx2a2ax52．dx＝x2a2Cx2x2a2xa253．x2a2dx＝xx2a2a2lnxx2a2C2254．(x2a2)3dx＝x(2x25a2)x2a23a4lnxx2a2C8855．xx2a2dx＝1(x2a2)3C356．x2x2a2dx＝x(2x2a2)x2a2a4lnxx2a2C8857．x2a2dx＝x2a2aarccosaCxx58．x2a2dx＝x2a2lnxx2a2Cx2x（八）含有a2x2(a0)的積分59．dxx2＝arcsinxCa2a60．dx＝xC(a2a2x2x2)3a261．xx2dx＝a2x2Ca262．xdx＝1C(a2a2x2x2)363．x2x2dx＝xa2x2a2arcsinxCa222a64．x2dx＝a2xx2arcsinxC(a2x2)3a65．dx＝1aa2x2Cxa2alnxx266．dx＝a2x2Cx2a2a2xx267．a(chǎn)2x2dx＝xa2x2a2arcsinxC22a68．(a2x2)3dx＝x(5a22x2)a2x23a4arcsinxC88a69．xa2x2dx＝1(a2x2)3C370．x2a22dx＝x22)a2x2a4xCx(2xa8arcsin8a71．a(chǎn)2xx2dx＝a2x2alnaa2x2Cx72．a(chǎn)2x2dx＝a2x2arcsinxCx2xa（九）含有ax2bxc(a0)的積分73．a(chǎn)x2dxc＝1ln2axb2aax2bxcCbxa74．a(chǎn)x2bxcdx＝2axbax2bxc4a75．a(chǎn)x2xcdx＝1ax2bxcbxa76．77．78．

dx＝1arcsin2axbCcbxax2ab24accbxax2dx＝2axbcbxax2b24acarcsin2axbC4a8a3b24accxdx＝1cbxax22barcsin2axbCbxax2aa3b24acxa(xa)(bx)的積分（十）含有或xb79．80．81．82．

xadx＝(xb)xa(ba)ln(xaxb)Cxbxbxadx＝(xb)xa(ba)arcsinxaCbxbxbxdx＝2arcsinxa(ab)(xa)(bx)bCx(xa)(bx)dx＝2xab(xa)(bx)(ba)2arcsinxaC44bx（十一）含有三角函數(shù)的積分83．sinxdx＝cosxC84．cosxdx＝sinxC85．tanxdx＝lncosxC86．cotxdx＝lnsinxC87．secxdx＝lntan(x)C＝lnsecxtanxC4288．cscxdx＝lntanxC＝lncscxcotxC289．sec2xdx＝tanxC90．csc2xdx＝cotxC91．secxtanxdx＝secxC92．cscxcotxdx＝cscxC93．sin2xdx＝x1sin2xC2494．cos2xdx＝x1sin2xC2495．sinn96．cosn

xdx＝1sinn1xcosxn1sinn2xdxnnxdx＝1cosn1xsinxn1cosn2xdxnn97．

dx

1cosxn2dx＝nsin98．cosn99．cosm

xn1sinn1xn1sinn2x＝11sinxn2dx2xxncosn1xn1cosnxsinnd＝1m1xsinn1xm1m2xsinnxdxxxmncosmncos＝m1cosm1xsinn1xn1cosmxsinn2xdxnmn100．1b)x1b)xCsinaxcosbxdx＝cos(a2(acos(a2(ab)b)101．1sin(ab)x1sin(ab)xCsinaxsinbxdx＝2(a2(ab)b)102．1sin(ab)x1sin(ab)xCcosaxcosbxdx＝2(a2(ab)b)dx2atanxbarctan2C(a2b2)103．＝b2a2absinxa2b2dx1atanxbb2a2104．ln2C(a2b2)＝absinxb2a2atanxbb2a22105．dx2abarctan(abx(a2b2)bcosx＝babatan)Caab2dx1abtanxab106．ln2baC(a2b2)bcosx＝bbaaatanxab2ba107．22dx22x＝1arctan(btanx)Cacosxbsinaba108．dxb2sin2x＝1lnbtanxaCa2cos2x2abbtanxa109．xsinaxdx＝12sinax1xcosaxCaa110．x2sinaxdx＝1x2cosaxxcosaxdx＝1a1111．cosaxa2a112．x2cosaxdx＝1x2sinaxa（十二）含有反三角函數(shù)的積分（此中

22xsinaxaxsinaxC2a2xcosaxa0）

23cosaxC2sinaxC113．a(chǎn)rcsinxd＝xarcsinxa2x2Caa114．xarcsinxdx＝(x2a2)arcsinxxa2x2Ca24a4115．x2xx3x1(x22a2)a2x2Carcsindx＝arcsin9a3a116．a(chǎn)rccosxdx＝xa2x2Caa117．xarccosxdx＝(x2a2)arccosxxa2x2Ca24a4118．x2arccosxdx＝x3arccosx1(x22a2)a2x2Ca3a9arctanxdxa22xarctanln(ax)C119．＝aa2x＝1xa120．a(chǎn)rctand22a22a121．x2arctanxdx＝x3arctanxax2a3ln(a2x2)Ca3a66（十三）含有指數(shù)函數(shù)的積分122．a(chǎn)xdx＝1axClna123．eaxdx＝1eaxCa124．xeaxdx＝1(ax1)eaxCa2125．xneaxdx＝1xneaxnxn1eaxdxaa126．xaxd＝xax12axCxlna(lna)127．xnaxdx＝1xnaxnxn1axdxlna1lna128．eaxsinbxdx＝a2b2eax(asinbxbcosbx)C1129．eaxcosbxdx＝2b2eax(bsinbxacosbx)Ca130．eaxsinnbxdx＝a2122eaxsinn1bx(asinbxbn131．eaxcosnbxdx＝a2122eaxcosn1bx(acosbxbn

nbcosbx)nbsinbx)（十四）含有對(duì)數(shù)函數(shù)的積分132．lnxdx＝xlnxxC133．dx＝lnlnxCxlnx11134．xnlndx＝xn1)Cn1n1(lnx)ndx＝x(lnx)nn1135．n(lnx)dx136．xm(lnx)ndx＝11xm1(lnx)nnxm(lnx)n1dxmm1（十五）含有雙曲函數(shù)的積分137．shxdx＝chxC138．chxdx＝shxC139．thxdx＝lnchxC140．sh2xdx＝x1sh2xC24141．ch2xdx＝x1sh2x