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1、常用积分公式（一）含有axb的积分（a0)axdx1,-=lnbaax2.(axb)dx(axb)1)3.dxaxbblnaxb)C4.2x.dxaxbb)22b(axb)b2lnax5.dxx(axb)1lnbax6.dxx2(axb)1bx7.xdx(axb)1=(lna8.2x(axb)2dx9.axaxbbaxb)1(axa2blnaxbb2axb)dx772x(axb)b(axb)1-2lnb2axb（二）含有“axb的积分10.11.12.Tax_dx=7(axb)33axx=x2.axbdx2(3ax2b15a:(15a2x212abx8b2).,(axb)3105a)(axb)
2、3C.一x,213.dx=一(ax2b)JaxbC、axb3a214.dxaxb2222(3ax4abx8b).axbC15a15.dxxyaxb*nVbvaxbVb2一arctanbaxb16.dxx2axbaxbbxa2b17.2axb18.axb19.20.(四)22.23.24.25.26.C(b0)C(b0)dxx.axbdxxvaxbdxx、ax含有x2a2的积分dx2a(x2=arctan-aadxa2)、2(n1)a2(x22、n1a)2n32(n1)a2dx二22、n1(xa)dx2xi含有2ax2axdxbx2ax-dxb2x2_-axb|xalxalCln2ab(a0)的
3、积分-arctanbx:、ax.、b(b0)2.顷dx=dxx(ax2b)ax(b0)12a,2lnaxln2bdxax2b2x2axdxx2(ax2b)bxbdxax2b27.28.(五)29.30.(六)31.32.33.34.35.36.37.38.39.dxa,2=lnx(axb)2bdx22(axb)含有ax22.axb2x_22b(axb)12bbxdxax2bxcx.2dxax成c含有、.x2dx.=22.xadxc(a0)的积分24acb2ln,b24aclnax2a2/a(a.xarsh-a=Tdx=adx23a)2x22.xadx=dx23a)x、x2a2dx222xxad
4、x2-axbx2axbcarctan-C、4acb2axbb24ac0)的积分2ax4acbxb2adx2axC1=ln(x,x2a2C(b2C(b2bxca2)Cx212aa2xx22ax2a2anx,x22aC2ax2.x22aJ22ln(x1Cadxln(xx22ln(x、x2a2)、x2a2)4ac)4ac)40.41.OVdx=-(2x285a2)./32z、-aln(xxa)C842.a2)3Cx2J_a2dx=(2x28a2)、x2a24ln(x、x2a2)C822243.dx=Tx2_a2aInCxxx2ax2a2J广44.2dx=ln(xVxa)Cxx(七)含有Jx2a2(a
5、0)的积分45.dxx.一archx2SaC1=lnxC46.dx,(x2a2)348.dx=49.2a.Inx2C50.x222a.xa47.:：2dx=Jx2a2Cxadx1dx_Jx2a2x2x2a2/x=arccosa53.Jx2a2dx=x0x2a22aIn2x/22vxaC54.*a2)3dx=-(2x285a2)4234a-a8In/22xvxa52.55.-221xVxadx=_356./dx弋(2x2a2hATW4In8xx258.,x22axx22r57.xInx59.60.62.63.64.65.66.含有a2dxa2x2dxx2(a0)的积分=arcsinCa.(a2x
6、2)3x2.：=22a.axx,ldx=22axdx.(a2x2)32x顼jdx=22ax2-=xdx(a2x2)3dxx、a2x2dx222x、ax67.Ja2x2dx=/22、3,68.(ax)d69.x、a2x2dx70.22.ax1222x2ax一arcsin2aarcsinCa22ax2ax2ax一arcsin2ax22、2=(5a2x)-a8=(a23x2)3C3a4arcsin8ax2x=(2x284a.xarcsin一C8a22旦X=40T7a72.x2Xa2J=arcsin-Ca(九)含有Jax2bxc(a0)的积分73.dxax2bxcIn2axb2焰bxcC74.ax2b
7、xcdx2axb-2:axbx4a4acb2.-In2ax8a3b2a;ax2bxx12；75.,2=dx=Vaxbxc.axbxca76.dx.cbxax22ax如arcsinb2、a、ax2bxc2axb.b24ac77.cbxax2dx22axb2b4ac.2axb八、cbxaxarcsinC4a8.a3、b24ac78.-2x=.cbxax2b_一一2axb八:arcsinC2、a3、.b24ac(十)含有J或彳xa)(bx)的积分80.(ba)ln(JxaJxb)C82.81.b),xabx=2arcsin(bC(ab).(xa)(bx)dx2xa4a)arcsinCbJ(xa)(b
8、x)(ba)2一.xa八arcsin.C4bx（十一）含有三角函数的积分83.sinxdx=cosxC84.cosxdx=sinxC85.tanxdx=86.87.88.sinxCtan(一-)42C.xtan2C=lnCIncosxIncotxdx=In=Insecxtanxsecxdx=Incscxdx=cscxcotxC89.2secxdx=tanxC90.2cscxdxcotxCsecxtanxdx=secxC92.98.2sinxdx=x214sin2xC2cosxdx=x214sin2xC_nsinxdx=1n1nsinxcosxnnncosxdx=1n1；n1cosxsinxnn
9、dx=1cosxn2-nsinxn1一_n1sinxn1dx1sinxn2C93.94.195.96.97.cscxcotxdx=cscxn2cosxdxcosnxn1cosn1xn2sinxdxdx一一n2sinxdxdn2n1cosx99.1m_._ncosxsinxdx=mm1.n1m1cosxsinxnmn1m1n1cosxsinxmn100.sinaxcosbxdx=1cos(ab)x2(ab)(ab)mcos2.nxsinxdxmn2cosxsinxdxmn1，、八cos(ab)xC2(ab)101.sinaxsinbxdx=1sin(ab)x2(ab)1.，、八sin(ab)xC
10、2(ab)102.cosaxcosbxdx=1sin(ab)x2(ab)1.，、sin(ab)x2(ab)103.dxabsinx2.a2.xatan=arctan2b2b-C(a2b2)104.dxabsinxatan2batan2b言a2b2C*a(a2b2)105.dxabcosx2abab_一,ab一x、arctan(,tan)ab.ab2(a2b2)106.dxabcosx1baabab107.dx_22.2.2acosxbsinxCC1arctan(tanx)ab108.dx22T22acosxbsinxbtanxabtanxaaCn2ab109.xsinaxdx=sinaxa11
11、0.x2sinaxdx=12xcosaxa1-xcosaxCa2xsinaxa2rcosax3a111.112.一.1一xcosaxdx=cosaxax2cosaxdx=1x2sinaxa含有反三角函数的积分(其中1xsinaxa2xcosaxaa0)113.x.x2arcsin-dx=xarcsinVaa114.115.116.(axA,Xxarcsindx=(a22a、)arcsin432_x,Xxxarcsin-dx=arcsin一a31/29(xxxarccosdx=xarccos-2.八sinaxCaa2x2C2a2)、.a2x2C117.2xxxarccosdx=(a22ax)ar
12、ccos一4x2-.a4118.x2arccosdxa3x=arccos39(x22a2)、.ax2C119.Xx,xxarctandx=xarctana120.aX1,2xarctandx=-(aa2a2ln(a2x2)C122.123.124.125.126.127.128.129.130.131.（十四）132.133.2xx)arctan3x2arctandx=arctana含有指数函数的积分axdx=axClnaeaxdx=1eaxCa_ax1xedx=(axanax1naxxedx=xeaxaxdx=Inaxnaxdx=xlna37ln(a2x2)Cax1)e1ax,edxa(lna)Inan1x.xadxeaxsinbxdx=22a2b21ax一e(asinbxbcosbx)eaxcosbxdx=a_axnesinbxdx=a)eax(bsinbxacosbx)b21ax,22ebnsinn1bx(asinbxnbcosbx)，2n(n1)b
