高等数学积分公式大全

常用积分公式（一）含有axb的积分(a0）１.dx＝1lnaxbCaxba2．(axb)dx＝1(axb)1C（1）a(1)3.xdx=12(axbblnaxb)Caxba4.x2dx＝11(axb)22b(axb)b2lnaxbCaxba325．dx＝1lnaxbCx(axb)bx6．2dxb)＝1a2lnaxbCx(axbxbxx2dx=1axbb)C7．(axb)a2(lnaxb８．x22dx＝13(axb2blnaxbb2)C(axb)aaxb9．dx11axbCx(axb)2＝b(axb)b2lnx(二)含有axb的积分1０．axbdx＝2(axb)3C3a１１．xaxbdx=22(3ax2b)(axb)3C15a12．x2axbdx＝23(15a2x212abx8b2)(axb)3C105a13.xbd=2(ax2b)axbCaxx3a214．x2bdx=2(3a2x24abx8b2)axbCax15a31lnaxbbC(b0)dxbaxbb１5．＝xaxb2arctanaxbC(b0)bb16.dx＝axbadxx2axbx2bxaxbb17．axbdx＝2axbbxdxbxax18．axbdx=axbaxdxbx2x2ax(三)含有x2a2的积分1９．dx2=1arctanxCx2aaa20．dx＝x2n3dx(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n121．dx=1xaCx2a2lna2ax(四)含有ax2b(a0)的积分1arctanaxC(b0)２2．dx=abbax21axbblnC(b0)2axbab２３.axxdx＝1lnax2bC2b2a２４．x2xbdxax2bdx=aaax2b２5.dxb)=1lnx2bCx(ax22bax2２6．dx=1adxx2(ax2b)bxbax2b２7.dx＝a2lnax2b1C3(ax22bx22xb)2bx28．dx＝x1dx(ax2b)22b(ax2b)2bax2b(五）含有ax2bxc(a0)的积分2arctan2axbC(b24ac)２9．dx＝4acb24acb2ax2bxc12axbb24acln(b24ac)b22axbb2C4ac4ac30．ax2xdx=1lnax2bxcbax2dxbxc2a2abxc(六）含有x2a2(a0)的积分31.dx=arshxC1＝ln(xx2a2)Cx2a2a3２．dx=xC(x2a2)3a2a2x2３3.x2xa2dx＝x2a2C３4．xdx＝1C(x2a2)3x2a235．x2dx＝xx2a2a2ln(xx2a2)Cx222a2３６.x2dx＝xln(xx2a2)C(x2a2)3x2a2３７．xdx＝1lnx2a2aCx2a2ax38.dxa2=x2a2Cx2x2a2x３9．x2a2dx＝xx2a2a2ln(xx2a2)C2240．(x2a2)3dx＝x(2x25a2)x2a23a4ln(xx2a2)Ca2dx＝18841.xx2(x2a2)3C3４2.x2x2a2dx＝x(2x2a2)x2a2a4ln(xx2a2)C884３.x2a2dx=x2a2alnx2a2aCxx４４．x2a2dx＝x2xa2ln(xx2a2)Cx2(七）含有x2a2(a0)的积分45.xdx=xarchxC1=lnxx2a2C2a2xa4６.dx=xC(x2a2x2a2a2)347.x2xdx＝x2a2Ca248.xdx＝1C(x2x2a2a2)34９．x2dx=xx2a2a2lnxx2a2Cx2a2225０．x2dx＝xlnxx2a2C(x2a2)3x2a251．xdxa2＝1arccosaCx2ax52.dx＝x2a2Cx2a2xx2a253.x2a2dx=xx2a2a2lnxx2a2C2254．(x2a2)3dx＝x(2x25a2)x2a23a4lnxx2a2C885５．xx2a2dx＝1(x2a2)3C35６.x2x2a2dx=x(2x2a2)x2a2a4lnxx2a2C8857．x2a2dx=x2a2aarccosaCxx58.x2a2dx＝x2a2lnxx2a2Cx2x(八)含有a2x2(a0)的积分5９．dxx2＝arcsinxCa2a60.dx=xC(a2x2)3a2a2x2６１．xdx＝a2x2Ca2x26２.xdx=1C(a2x2)3a2x263．x2dx＝xa2x2a2arcsinxCa2x222a6４.x2x2)3dx=a2xx2arcsinxC(a2a65．dxx2＝1lnaa2x2Cxa2ax66．dx=a2x2x2a2a2xCx267.a2x2dx=xa2x2a2arcsinxC22a68．(a2x2)3dx＝x(5a22x2)a2x23a4arcsinxC88a69.xa2x2dx＝1(a2x2)3C370．x2a2x2dx＝x(2x2a2)a2x2a4arcsinxC88a71．a2x2a2x2alnaa2x2Cxdx＝x７2.a2x2a2x2xx2dx=xarcsinaC(九）含有ax2bxc(a0)的积分73.ax2dx＝1ln2axb2aax2bxcCbxca74．ax2bxcdx=2axbax2bxc4a4acb2ln2axb2a2bxcC8a3ax７5.ax2xdx＝1ax2bxcbxca2bln2axb2aax2bxcCa376.dx＝1arcsin2axbCcbxax2ab24ac７7.cbxax2dx＝2axbcbxax2b24acarcsin2axbC4a8a3b24ac78.xdx=1cbxax2barcsin2axbCcbxax2a2a3b24ac(十)含有xa或(xa)(bx)的积分xb７9.xadx=(xb)xa(ba)ln(xaxb)Cxbxb8０．xadx＝(xb)xa(ba)arcsinxaCbxbxbx81．dx=2arcsinxaC(ab)(xa)(bx)bx82．(xa)(bx)dx=2xab(xa)(bx)(ba)2arcsinxaC44bx(ab)（十一）含有三角函数的积分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.sinnxdx＝1sinn1xcosxn1sinn2xdxnn96．cosnxdx＝1cosn1xsinxn1cosn2xdxnn97.dx1cosxn2dx=sinn1xn1sinn2xsinnxn198.dx＝1sinxn2dxcosnx1cosn1xn1cosn2xn99.cosmxsinnxdx＝1cosm1xsinn1xm1cosm2xsinnxdxmnmn=1cosm1xsinn1xn1cosmxsinn2xdxmnmn100.sinaxcosbxdx＝1cos(ab)x1cos(ab)xC2(ab)2(ab)1０１．sinaxsinbxdx＝1sin(ab)x12(asin(ab)xCb)2(ab)102．cosaxcosbxdx＝1sin(ab)x12(ab)sin(ab)xC2(ab)dx2atanxb１03．a=arctan2C(a2b2)bsinxa2b2a2b2dx1atanxbb2a2１0４．=2(a2b2)absinxb2a2lnatanxbb2a2C2０．dx=2abarctan(abtanx)C(a2b2)15abcosxababab2dx1abtanxab106.＝2baC(a2b2)aabblnbcosxatanxab2ba１07．dx=1ba2cos2xb2sin2xarctan(tanx)Caba10８．dx＝1lnbtanxaCa2cos2xb2sin2x2abbtanxa109．xsinaxdx=11Ca2sinaxxcosaxa1１０．x2sinaxdx＝1x2cosax2xsinax2cosaxCaa2a3111.xcosaxdx=12cosax1xsinaxCaa112.x2cosaxdx＝1x2sinax22xcosax23sinaxCaaa(十二）含有反三角函数的积分(其中a0)113.arcsinxdx=xarcsinxa2x2Caa１14．xarcsinxdx＝(x2a2)arcsinxxa2x2Ca24a4１15．x2arcsinxdx＝x3arcsinx1(x22a2)a2x2Ca3a9116.arccosxdx=xarccosxa2x2Caa１1７．xarccosxd＝x2a2xxa2x2Cax(4)arccos42a１1８.x2arccosxdx=x3arccosx1(x22a2)a2x2Ca3a9１19．arctanxdx＝xarctanxaln(a2x2)Caa2１２0．xarctanxdx=1(a2x2)arctanxaxCa2a2１2１．x2arctanxdx＝x3arctanxax2a3ln(a2x2)Ca3a66(十三)含有指数函数的积分12２．axdx＝1axClna1２3．eaxdx=1eaxCa１24.xeaxdx=1(ax1)eaxCa2１25.xneaxd1neaxnxn1axxaedxa12６.xaxdx=xax12axClna(lna)127．xnaxdx＝1xnaxnxn1axdxlna1lna1２8．eaxsinbxdx=a22eax(asinbxbcosbx)Cb129．eaxcosbxdx＝a21b2eax(bsinbxacosbx)C30．(na2

eaxsinnbxdx=a212n2eaxsinn1bx(asinbxnbcosbx)b1)b2axsinn2bxdxb2n2e131．eaxcosnbxdx=a2122eaxcosn1bx(acosbxnbsinbx)bnn(n1)b2axn2a2b2n2ecosbxdx(十四）含有对数函数的积分13２．lnxdx＝xlnxxC１3３.dx=lnlnxCxlnx11１３4.xnlnxdx=xn1(lnx)Cn1n1１35.(ln)ndnn1xn(lnx)dxx=x(lnx)1３6.xm(lnx)ndx=11xm1(lnx)nnxm(lnx)n1dxmm1(十五)含有双曲函数的积分137．shxdx＝chxC1３8.chxdx=shxC139．thxdx＝lnchxC140．sh2xdx＝x1sh2xC24141．ch2x