积分公式大全文档格式.docx
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积分公式大全积分公式大全常用积分公式4.-一X=-A3卩(ax+b)2-2b(ax+b)+b2lnax+b+Caxba2fdx=1lnax+b+CJlnx(ax+b)bX5.
(二)含有.axb的积分10.Jax+bdx=2J(a+b)3+CL3a11.x、.axbdx=飞(3ax-2b)(axb)3C15a12.2.axbdx=2(15a22-12abx8b2).(axb)3C105a13.JXd=-2y(ax-2b)0x+b+CaXb3a15.dxXMaxb2、-baxb:
@#@-b(b0)(b:
@#@0).axb-.b.aXbb(三)含有x2-a2的积分.dx1X19.-22=arctan一Cx2+a2aaFdxX2n3FdX20=22、n2,22、n42.,22、n_J21.dx=X-a12aInra(Xa)2(n-1)a(Xa)2(n-1)a(Xa)rdxJaretaab晶+c(bA0)IaXHbJln2-abC(beO)+V-b2(四)含有axb(a0)的积分22.X1223.12dx=lnax+b+Cax+b2a24.X2dXaxbdx2axb25.dxx(ax2b)丄In2b2Xax2b26.ILdxx2(ax2b)1bx27.dx3x3(ax2b)2b228.29.30.(A)31.32.33.dxbax2bax2bx2rdx222I(axb)2b(axb)2bax2bdx2-含有ax2bxc(a0)的积分ILdxax2bxC-arctan2JC4ac-b2In、b2-4ac戶dx=axbxC2a.4ac_b22axb-,b2-4ac2axb.b2-4acdx2_2(b:
@#@4ac)C(b24ac)Ina2+bx+c.22aax“+bx+c含有.X2a2(a0)的积分dx-x2a2dx.(x2a2)3arshC1=In(x.x2a2)CaaCxa2dx=x2a2C1dx=-=C.(x2a2)3x2a235.36.37.38.39.40.41.42.43.44.45.46.47.,x2aXr22a22=dx=.xaln(x.Xa)C22222、3dX=2flnXX)C(X2a2)X-adxxix2a2a1.x2a2-a=InC2dxx2,x2a2rdx=?
@#@|n(x.rCj(x2a2)3dx=-(2x25a2).x2a2-3a4ln(x8Cj2+a2dx=L322、3x2x2a2dx=xs2-2(2xa)x2a2ln(Xx2XdX=JK+aln迂aa+C222a2)Ca2)C-dx三x2a2In(xx2a2)C含有.2-a2(a0)的积分dX22X-adX=archX+C1=InX+寸x2_a2+CX2-a2)322dX=2_a2C48.49.50.51.52.53.54.55.56.57.58.(A)59.60.PdX=-.212+C-a).x-aX*2jl22XrX-a2IXr2TaIdx=X-aIn2XIXx-a21=arccosaXdx222.,口dx=号lnC2X(X2_a2)3dx=xX(225a2)、2_a8ILj2-a2dx=IJ(X2_a2)3+CL3+3a4nX8+4a2+Cx2、FdX=j22-血F4-lnx+Px-a+C8Vx2-a2-aarccosa22.X-a.dx=X-22X-aI2dx=X22X-a+Inx+2-a2含有.a2-x2(a0)的积分dxa2=2X=arcsnCadx.(a2-X2)3-+c222aa-X61.62.63.64.65.66.67.68.69.70.71.72.(九)73.J/22dx=-da2x2+cI.dx=.(a2-x2)31+C.a2-x2,a-X2f,Xdx=JIl22.a-X2X22ax.a-XarcsInC22aIX.Xdx1I=lna2-x2a;@#@22ya-X+cdxx2.a2-x22Ja2-x2dx=XJa2_x2+arcsin+C22a!
@#@;@#@(a2_x2)3dx=;@#@(5a2_2x2).a2_x2Jxa7dx=一3JU-a4arcsinXC8a42-a2-2dx=(22-a2)、a2-2-arcsinC88a22adx=、a2x2alnXJ22a_aXC_22-22axa-xXC2dx=arcsInCXXa含有-ax2bxc(a0)的积分dx=丄.ax2bxcIaIn2ax+b+2寸aJax2+bx+c+C74.22axb-2axbxCdX=axbxC4a4ac-b18苛l2xb2aaxbxCC75.ax2bxCdx=丄Ja2+bx+ca22xb2a76.dxICbx-a212ax-barcsin:
@#@2+Ca,b4ac77.Cbx-ax2dx2ax一b2bx-a24ab24ac.2ax-barcsInC8ab24ac78.-dx-Cbx-axIJC+bx-a2a+Aarcsin卓+C2a3-b24ac或、.(xa)(bx)的积分81.82.79.80.含有*兰(X一b)JbWa)In(J(b-a)arcsinI-_+Cb-x(x-b)x-ab-xXa+x_b)+c,(x-a)(b-x)2arcsin+C(acb).(x-a)(b_x)d2xa-bX=(x-a)(b-x)+守arcsin后+C(a:
@#@b)(十一)含有三角函数的积分83.Sinxdx=-CoSXC84.85.86.87.88.89.90.91.92.93.94.95.96.97.98.99.100.CoSXdX=SinxCILtanXdX=Incosx+CcotXdX=lnSinx+CJTXsecxdx=lntan(上+)+C=lnsecx+tanx+C42Xfcscxdx=lntan+c=lncscxcotx+CL22SeCXdX=tanxCCSCXdX=-cotXCSeCXtanXdX=SeCXCCSCXcotXdX=-CSCXCsin12XdX=X1sin2xC242X1cosXdX=Sin2xCSinndXdX24cosnXdX1n4.n-1cosXSinxCOSnXdXnndx1cosxn-2dxnSinXn-1n4SinXn-1!
@#@n_2SinXdx1SinXn-2dxnn-1+n-2cosXcosXnJAcosXnTSinnXdX=-丄Sinn4xcosxn-nnI-mnIcosXSinXdXm-1n十cosXSinXI-m-2nIcosXSinXdXmn-2IcosXSinXdXSinaxCOSbXdX=-12(ab)cos(ab)x-12(a-b)cos(a-b)xSinaxSinbxdx=cosaxCOSbXdX=12(ab)12(ab)sin(ab)xsin(ab)x12(a-b)12(a-b)Sin(a-b)xSin(ab)XC101.102.103.104.105.106.107.108.109.110.111.112.Xatanb22arctanCab.a-b上叫I:
@#@二;@#@Xabtan2lb-afdxa2cos2Xb2sin2Xabfdx.222-2-acosx-bSinX丄ln2abbtanxabtanx-adxlnababcosx11XSinaxdx=2SinaxXcosaxCaa2.2+2+2+qXSinaxdx=Xcosax2xsInaX亍cosaxCaaa11XCOSaXdX=2cosaxXSinaxCaax2COSaXdX=1x2sinaxWxcosax-sinaxCaaa(十二)含有反三角函数的积分(其中a0)XXf22113.arcsindx=XarCSina-xCaa22,.XXa、.XX口2C114.XarCSIndx=()arcsna-XCa24a43dXX12222=arcsIn(X2a)、a-XC3a9118.x2arccos二dx=arccosx-l(x22a2),a2-x2Ca3a9119.arctanxdx=Xarctan-aln(a2x2)Caa2X122Xa120.XarCtandx=(aX)arctanXCa2a2332XXXa2a22121.Xarctandx=arctanXln(aX)Ca3a66(十三)含有指数函数的积分122.123.124.125.126.127.128.129.1axdx=axClnaax1axedx=eCaXeaXdX=-12(ax-1)eaxCanax1naxnn4ax.XedX=XeXedXaaLXaXdX=ax12axClna(lna)XnaXdX=1nXalnaeaxSinbxdx=eaxcosbxdx=nn4XIXadXlna1a2b2eax(asinbx-bcosbx)C1a2b2eax(bsinbxaCOSbX)C138.ChXdX=ShXC(十四)含有对数函数的积分139.thxdx=InChXC140.sh2xdx=-X-sh2xC24r2XI1丄141.ChXdX=sh2xC4(十六)定积分142.cosnxdx=Sinnxdx=0-.143.COSmXSinnxdx=0-二0,mn144.COSmXCOSnxdx=I,m=n-0,m=n145.SinmXSinnxdx=入:
@#@,m=n146.卩,Sinmxsinnxdx=COSmXCOSnxdx=LOI,m=n2147.InO2SinnXdX=02CoSnXdXInn-1In1n_2Inn-3.n-2n-3(n为大于1的正奇数),Ii=1n-2Jl422(n为正偶数)ln.X-ax2.,x2_a21m.n4丄n1cosXSinXmnmn