考研常用积分公式（范文篇）

考研常用积分公式(范文4

篇)以下是网友分享的关于考研常用积分公式的资料4篇,希望对您有所帮助，就爱阅读感谢您的支持。考研常用积分公式(1)(xN)/="ji-1(ax)/=axIna(ex)/=ex(log/lax)=xIna(Inx)/l=x(sinx)/=cosx(cosx)/="sinx(tanx)/=sec2x(cotx)/="csc2x(secx)/=secxtanx(escx)/="cscxcotx(arcsinx)/(arccosx)/=(larctanx)/(arccotx)/=-ll+x2(uv)//v+uv/Ll/.u/v-uv/\vU=v2Jkdx=kxxJl+1Jxjidx=p+lJdxx=lnxJdx1+x2=arctanx=arcsinxJcosxdx=sinxfsinxdx=-cosxJsec2xdx=tanxJcsc2xdx=-cotxJsecxtanxdx=secxJescxcotxdx=-cscxJexjaxdx=axIna\tanxdx="lncosxJcotxdx=lnsinx\secxdx=lnsecx+tanxfescxdx=lnescx-cotxJllx2+a2dx=aarctanxaJllx-x2-a2dx=2aInax+a=lnxx=arcsina等价无穷小(x—0)sinx〜xtanx〜xarcsinx〜xarctanx〜xln(l+x)~xex-1〜x121-cosx〜2xax-1〜xInaf(x)渐近线k=limx—>ooxb=limx—>ooFLf(x)-kx1Jy//曲率k=(31+y/2)21考研常用积分公式(2)//(x)氏1(arctanx)=(arccotx)/11+x2fcscxdx=-cotx22Jl=arcsinxa(a)x/=aIna=e/xsecxtanxdx=Jescxcotxdx=Jedx=eJadx=secx等价无穷小(x-0)-CSCXsinx〜xtanx〜x(e)x/(uv)lxIna/=UV4-UV(logax)(lnx)axarcsinx〜x(sinx)Jkdx=kxInaarctanx〜x=cosx(cosx)(tanx)(cotx)(secx)(cscx)/=-sinx=secx=-cscx=secxtanx=-cscxcotxxdx=xjl+1jl+1Jtanxdx="lncosxJcotxdx=Jsecxdx=InsinxInsecx+tanxln(l+x)~xe-1-xdx/J1+xxdx=lnx=arctanx1-cosx〜1222/JJescxdx=lnescx-cotx=arcsinx1〜x/JxJx12+a12dx=laarctanxaa-1〜xIna(arcsinx)Jcosxdx=sinx2-a2dx=12aInx-ax+a渐近线k=limf(x)XX—>00Jsinxdx=2-COSX(arccosx)/1Jsecxdx=tanxJ=lnx+b=limFLf(x)-kxlJx-oo最新下载(NewD)中国最大、最专业的学习资料下载站转载请保留本信息曲率k=//32d+y)/2最新下载(NewD)中国最大、最专业的学习资料下载站转载请保留本信息考研常用积分公式(3)当x—0时x-sinx〜tanx〜arcsinx〜arctanx〜ln(l+x)〜2.常用极限limnkx—>coax—>ooax-1Ina1+xb-1b(其中a>0,b#0)x〜ex-1x2-l-cosxx—1a〜(1+x)a2n11n=0,(a>l)limcnx—>oon!x—go=0,(c>0)limnqn,(qx—>oonlimn=1,(a>0)6.limlogannx—>qo=0,(a>l)Inlim9.limx0eliml+=ex—>oonx10.limsinxxx—>co=0)=lp+111.limlogaxx£X—>+oo=0,(a>l,e>0)np+112.lim(x—>qoIp+2p4-+npnp+12pp+113.lim(x—>ooIp+2p+-+npnp+lln+lln+2=212n114.lim(x—>coIp+3p4-+(2n-l)pnp+1X—>C0++••+=ln2P17.lim(1+x)=ex—01x16.limsinxxax-lxx—0=1=amlim20.lim22.lim24.lim26.lim28.x—>0=lna=1=1lim21.lim23.lim25.lim(H-a)ji-laarcsinxxln(l+x)xarctanxxmx—Ox-0=1=mn(n-m)21(1+mx)n-(1+nx)mx2mx—Ox70n-xx-0(mnr0)nn,laxmx—0+,(mn#0)xm-lxn-1mx—>1mnn,(m,n为自然数)，(m,n£Z)27.limx—1m1-xmn1-xnm-n2x一m29.若Xn(n=l,2...)收敛，则算数平均值的序列；n=X1+X2+Xn,(n=l,2)也收敛，且n1limxl+x2+…+xnnX一GO=limxnX一0030.若序歹l|Xn(n=l,2.・・)收敛，且Xn>0,则lim=limXnx—>coX—>00n31.若Xn>0(n=l,2...)且limXn+1x—>ooXnlim=limx—>00Xn+1x—>coXn32.若整序变量Yn-+oo,并且——至少是从某一项开始——在n增大时Yn亦增大，Yn+l>Yn,则n—>coYnlimXn=limXn-Xn-1n—>ooYn-Yn-1.常用符号.微分学基本公式.y=cdy=OIcosXl+2+-+n=n(n+l)212+22+…+n2=nn+1(2n+l)63.13+23+…+n3=(14-2+-n)24.a3±b3=a+b(a2Tab+b2)xn-1=x-1(xn-1+xn-2+“+x+1)xn-an=x-a(xn-1+axn-2+a2xn-3+-+a2x+an-1)7.xn+an=x+a[x2k-1-ax2k-2+…+(a2k-2x-a2k-1)]8.x-1=(++-+1)9.伯努利不等式(1+x)n>l+nx1+xll+x2•••1+xn>l+xl+x24-+xn|x-y|>|x-|y|||xy|>xyX+X1++Xn>X-X1++Xn13.n!ln+1n+ln)21114. 24132n-12nnna-Inl+al7.n-lkAk组合数公式Cn=k!n!k!n-k!kk-1Cm+n+l-Cmk=Cm+n+nn!k升F歹!J数公式An=n,n-1••…n-k+1=n-k!z6-1=z+1z-1z2+z+1z2-z+120.z6+1=z2+1(z4-z2+1)21.z4+1=z2++1(z2-+1).记号n!!表示自然数的连乘积，这些自然数不超过n,并且每两个数之间差2.例：7!!=135・78!!=2・4・6・8dx.4.6.8.y=xgdy=^ixji-ldxy=logaxdy=logaexy=axdy=axInadxy=sinxdy=cosxdxy=tanxdy=sec2xdx=dxy=cosxdy=-sinxdxy=cotxdy=-csc2xdx=Isinxdxy=secxdy=secxtanxdx11+xlch2xy=cscxdy=-cscxcotxdx12.y=arccosxdy=14.y=arccotxdy=-18.y=cthxdy=-11+xlsh2xy=arcsinxdy=13.y=arctanxdy=17.y=thxdy=dxdxy=shxdy=chxdxdxy=chxdy=shxdxdx6.不定积分表Odx=c3.x|idx=5.ll+xlji+lxaIdx=x+c4.dx=ln|x|+c.XJl+l+cdx=arctanx+caxIna=arcsinx+caxdx=+c8.sinxdx=-cosx+c10.14.16Isin2xlshxdxax+x2xdxa±x9.cosxdx=sinx+c11.Icosxdx=-cotx+cdx=tanx+c12.shxdx=chx+cdx=-cthx+c=arctan+c,(ar0)aa1x13.chxdx=shx+c15.2dx=thx+cchx17.=ln||+ca-x2aa-xxdx120.=±in|a2±x2|+c21arcsin+c,(a>0)a21.=ln|x++c±+cx222.dx=+arcsin+c,(a>0)7.三角学公式sin20cos20.基本关系sin0-csc0=13.tan0-cot0=15.sec20-tan20=17.tan0=sin0cos022.dx=±2xa222.cos0-sec0=14.sin20+cos20=16.esc20-cot20=18.cot0=cosOsin02.两角和与差的三角函数公式sina±p=sinacosp±cosasin03.tana±p=tana±tanpiTtanatanpcosa±p=cosacos0干sinasin04.cota±p=cotacotpTlcotp±cota3,倍角公式sin2a=2sinacosa=2tanal+tana1-tan2al+tanacos2a=cos2a-sin2a=2cos2a-l=l-2sin2a=tan2a=2tanal-tana=(sinal+cosa2.2cot2a=cot2a-12cotasin3a=3sina-4sin3a1.3.sin2=2a1-cosa2cos3a=4cos3a-3cosacos2=2a1+cosa2.半角公式=(tan2=2al-cosal+cosal-cosa2)sinacot2=2al+cosal-cosa=(l+cosa2sina=(sina1-cosa)2.和差化积公式.5.6.7.sina+sinp=2sinsina-sinp=2cosa+p2a+p2cossina-p2a-picosa+cosp=2cosa+p2cosa-p2cosa-cosp=-2sintana±tanp=±cota±cotp=±tana±cota+02sina—02sin(a±p)sinasinpcos(a邛)sinasinpcos(a邛)cosasinp6.积化和差公式1.2.3.sinasinp=-[cosa+0-cos(a-p)]cosacosp=[cosa+p+cos(a-p)]sinacosp=[sina+p+sin(a-p)]2211217.双曲函数的基本关系3.5.7.cosh2t-sinh2t=1coth2t=l+sinhx=2Isinh2t4.6.1-tanh2t=Icoshtsinh2x=2sinhxcoshx=2cosh2x-tanhxcoshx=ex+e-x2ex-e-x双曲余弦的反函数x1+tanxx=ln(y±y8.万能公式1.3.5.sinx=tanx=secx=2tan2.4.6.COSX=cotX=CSCX=1-tan2x1+tanx1-tan2x2tan2x1-tanx2tanl+tan2x1-tanxl+tan2x2tan考研常用积分公式（4）常用积分公式(一)含有ax+b的积分(a#0)dxllnax+b+CJ1.ax+b=alji+l(ax+b)+C(ax+b)dxa(ji+1)J2.= gxlx(ax+b-bInax+b)+C2Jax+ba3.=1F11x222(ax+b)-2b(ax+b)+bInax+b+Cx3IIJa2J4.ax+b=Ldxlax+b-Jx(ax+b)bInx+C5.=dxlaax+b-+ln+CJx2(ax+b)bxb2x6.=xlbx(Inax+b+)+C2J(ax+b)2aax+b7.=2x21bJ(ax+b)2xa3(ax+b-2bInax+b-ax+b)+C8.=dx11ax+b-In+CJx(ax+b)2b(ax+b)b2x9.=的积分CxlO.J2(3ax-2b+Cx211.J=15a12.Jx2222(15ax-12abx+8bCx3=105a132x(ax-2bC=3a214*22x(3a2x2-4abx+8b2+C3=15a(15.=+C(b>0)C(b16.Ja-2b17.bx=18.a+x2=22(三)含有x土a的积分dxlxarctan+C22Jx+aaa19.=x2n-3dxdx+f(x2+a2)n2(n-1)a2(x2+a2)n-12(n-1)a2J(x2+a2)n-120.=lx-adxIn+C22f21.x-a=2ax+a2(四)含有ax+b(a>0)的积分fdxJ222.ax+b=x+C(b>0)+C(bxlxInax2+b+C2J23.ax+b=2ax2xbdx-xf2J224.ax+b=aaax+blx2dxIn2+C2J2bax+b25.x(ax+b)=dxladxJx2(ax2+b)-bx-bJax2+b26.=ax2+bdxalln-+C\x3(ax2+b)2b222x2bx27.=xIdxdx+f(ax2+b)22b(ax2+b)2bfax2+b28.=2(五)含有ax+bx+c(a>0)的积分2+C(bax2+bx+c=+Cx30.Jax2+bx+cx1=2aInax2+bx+c-bdx2aJax2+bx+c(a>0)的积分31.=arshxa+Cl=ln(x++C+cx33.C34.Jx=+C235*xa22ln(x+C2x+ln(x++C36.=(b2>4ac)37.J1+Ca38+C=392a+ln(x++Cx240.Jx3422(2x+5aaln(x++Cx8=8Cx41.J42♦xj4xa22(2x+aln(x++Cx8=843*aaIn+Cxx=44■x+ln(x+C=(a>0)的积分45.Jxxarch+Cllnx+Ca=x=46.+C47x=+C49■J22xa21nx++C250.x+lnx++C51.Jlaarccos+Cax52.+C=53■2aInx++Cx254.Jx3422(2x-5aaInx++Cx8=8Cx55.Jx56.J4xa22(2x-aInx++Cx88=.Jaaarccos+Cxx=58*x+lnx++C=(a>0)的积分59.=arcsinx+Ca60.J+Cx61.=C62.Jx+C63■J2axxarcsin+C2a=22x-arcsin64.x+Ca.Jla-In+Cax66+C=672axarcsin+Cx2a68.Jx34x22(5a-2xaarcsin+Cx88a=Cx69.J=70.xJ4xax22(2x-aarcsin+Cx88a=71x=a+C72xx-arcsin+Ca=(a>0)的积分73・J2ax+b++Cx74.J+22ax+b++C752ax+b++C76.J=+c77.J2+Cx=+C=78x(x-b(b-a=+C80.x(x-b(b-aC=C81.(a82.J(b-a)2Cx4(a(十一)含有三角函数的积分sinxdx-cosx+CJ83.=cosxdxsinx+CJ84.=tanxdx-Incosx+CJ85.=86.Jcotxdx=lnsinx+C7txIntan(+)+CsecxdxInsecx+tanx+Cf4287.==escxdxJ88.=2secJxdxIntanx+CInescx-cotx+C2=89.=tanx+C=-cotx+C90.2cscfxdxsecxtanxdxsecx+CJ91.=escxcotxdx-escx+CJ92.=x1-sin2x+CsinxdxJ93.=242x1+sin2x+Ccosxdxf2494.=2In-In-1-sinxcosx+sinn-2xdxsinxdxJn95.J=nnIn-In-Icosxsinx+cosn-2xdxcosxdxJn96.J=nndxIcosxn-2dx-+n-1JnJn-297.sinx=n-Isinxn-IsinxdxIsinxn-2dx+11-1JnJn-298.cosx=n-Icosxn-IcosxIm-Im-In+lcosxsinx+cosm-2xsinnxdxcosxsinxdxJm+n99.J=m+nmnIn-Imn-2cosm+lxsinn-lx+cosxsinxdxJm+n=m+n-llcos(a+b)x-cos(a-b)x+CsinaxcosbxdxJ2(a+b)2(a-b)100.=-llsin(a+b)x+sin(a-b)x+CsinaxsinbxdxJ2(a+b)2(a-b)101.=-llsin(a+b)x+sin(a-b)x+Ccosaxcosbxdx2(a+b)J2(a-b)102.=x+b+CdxJa+bsinxatan(a2>b2)+CdxJa+bsinx=(a2xdx)+CJ2105.a+bcosxdxJ106.a+bcosx=+C(a2>b2)(a2dxlbarctan(tanx)+C2222Jacosx+bsinxaba107.=lbtanx+adxIn+C2222J2abbtanx-a108.acosx-bsinx=Hsinax-xcosax+Cxsinaxdx2ja109.=a1222xcosax+xsinax+cosax+Cxsinaxdx23Jaa110.=a2-llcosax+xsinax+Cxcosaxdx2Jaa111.=1222xsinax+xcosax-sinax+Cxcosaxdx23faaa112.=2(十二)含有反三角函数的积分(其中a>0)xxarcsinxxarcsin++CJa=a113.x2a2xxxarcsindx(-)arcsin+Cja=24a114.x3x12xxarcsinxarcsin+(x+2a2CJa=3a9115.2xxarccosxxarccos-C]aa116.=x2a2xxxarccosx(-)arccos-Cja=24a117.x3x12x2xarccosdxarccos-(x+2aCJa=3a9118.2xxa22arctandxxarctan-ln(a+x)+CJa=a2119.x12xa2xarctandx(a+x)arctan-x+CJa=2a2120.x3xa2a3xxarctandxarctan-x+ln(a2+x2)+CJa=3a66121.2(十三)含有指数函数的积分lxa+CadxJina122.=xlaxe+CedxJa123.=axl(ax-l)eax+Cxedx2124.J=aax125.naxxJedxInaxnn-laxxe-Jxedxa=axxlxa-a+C2xadxInaJ(Ina)126.=xInxnxa-xn-laxdxxadxjlna127.J=lnanxleax(asinbx-bcosbx)+Cesinbxdx22128.J=a+baxlaxe(bsinbx+acosbx