微积分常用公式

1.常用等价无穷小当x→0时x~sinx~tanx~arcsinx~arctanx~ln(1+x)~ax−1lna~(1+x)b−1b(其中a0,𝑏≠0)x~ex−112x2~1−cosx1nx~√1+xn−1α~(1+x)α2.常用极限1.limx→∞nan=0,(a1)2.limx→∞cnn!=0,(c0)3.limx→∞nqn,(|q|1)4.limx→∞√an=1,(a0)5.limx→∞√nn=16.limx→∞logann=0,(a1)7.limx→∞1√n!n=08.limx→∞(1+1n)n=e9.limx→∞n√n!n=e10.limx→∞nxx=011.limx→+∞logaxxε=0,(a1,𝜀0)12.limx→∞(1p+2p+⋯+npnp+1)=1p+113.limx→∞(1p+2p+⋯+npnp+1−np+1)=1214.limx→∞(1p+p+⋯+(2n−1)pnp+1)=2pp+115.limx→∞(1n+1+1n+2+⋯+12n)=ln16.limx→nxx=117.limx→(1+x)1x=e18.limx→ax−1x=lna19.limx→(1+a)μ−1a=μ20.limx→ln(1+x)x=121.limx→arcnxx=122.limx→arctanxx=123.limx→(1+mx)n−(1+nx)mx2=12mn(n−m)24.limx→√1+αxm−√1+βxnx=αm−βn,(mn≠0)25.limx→√1+αxm∙√1+βxn−1xαm+βn，（mn≠0）26.limx→1xm−1xn−1=mn,(m,n为自然数)27.limx→1(m1−xm−n1−xn)=m−n228.limx→1√xm−1√xn−1=nm,(m,n∈Z)29.若Xn（n=1,2…）收敛,则算数平均值的序列𝜁𝑛=1n(X1+X+⋯Xn),(n=1,⋯)也收敛，且limx→∞x1+x2+⋯+xnn=limx→∞xn30.若序列Xn(n=1,2…)收敛，且Xn0,则limx→∞√x1+x+⋯+xnn=limx→∞Xn31.若Xn0(n=1,2…)且limx→∞Xn+1Xn存在，则limx→∞√Xnn=limx→∞Xn+1Xn32.若整序变量Yn→+∞,并且——至少是从某一项开始——在n增大时Yn亦增大，Yn+1Yn，则limn→∞XnYn=limn→∞Xn−Xn−1Yn−Yn−13.常用公式及不等式1.1++⋯+n=n(n+1)22.12+2+⋯+n2=n(n+1)(2n+1)63.1++⋯+n=(1++⋯n)24.a±b=(a+b)(a2∓ab+b2)5.xn−1=(x−1)(xn−1+xn−2+⋯+x+1)6.xn−an=(x−a)(xn−1+axn−2+a2xn−+⋯+a2x+an−1)7.xn+an=(x+a)[(x2k−1−ax2k−2)+⋯+(a2k−2x−a2k−1)]8.x−1=(√xn−1n+√xn−2n+⋯+1)9.伯努利不等式(1+x)n≥1+nx(1+x1)(1+x)∙∙∙(1+xn)≥1+x1+x+⋯+xn10.|x−y|≥||x|−|y||11.|xy|≥xy12.|X+X1+⋯+Xn|≥|X|−(|X1|+⋯+|Xn|)13.n!(n+12)n14.12∙4∙⋯∙2n−12n1√2n+115.1n+1ln(1+1n)1n16.1+aea17.√an−1a−1n18.组合数公式Ckn=Ank!=n!k!(n−k)!Ckm+n+1−Ck𝑚+𝑛=𝐶𝑘−1𝑚+𝑛排列数公式Akn=n∙(n−1)∙⋯∙(n−k+1)=n!(n−k)!19.z6−1=(z+1)(z−1)(z2+z+1)(z2−z)+120.z6+1=(z2+1)(z4−z2+1)21.z4+1=(z2+√z+1)(z2−√z+1)4.常用符号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！！=1∙3∙5∙78‼=∙4∙6∙85.微分学基本公式1.y=cdy=02.y=xμdy=μxμ−1dx3.y=axdy=axlnadx4.y=logaxdy=logaexdx5.y=sinxdy=cosxdx6.y=cosxdy=−sinxdx7.y=tanxdy=sec2xdx=1co2xdx8.y=cotxdy=−csc2xdx=1n2xdx9.y=secxdy=secxtanxdx10.y=cscxdy=−cscxcotxdx11.y=arcsinxdy=1√1−x2dx12.y=arccosxdy=−1√1−x2dx13.y=arctanxdy=11+x2dx14.y=arccotxdy=−11+x2dx15.y=shxdy=chxdx16.y=chxdy=shxdx17.y=thxdy=1ch2xdx18.y=cthxdy=−1h2xdx6.不定积分表1.∫0dx=c2.∫1dx=x+c3.∫xμdx=1μ+1xμ+1+c4.∫1xdx=ln|x|+c.5.∫11+x2dx=arctanx+c6.∫1√1−x2dx=arcsinx+c7.∫axdx=axlna+c8.∫sinxdx=−cosx+c9.∫cosxdx=sinx+c10.∫1n2xdx=−cotx+c11.∫1co2xdx=tanx+c12.∫shxdx=chx+c13.∫chxdx=shx+c14.∫1h2xdx=−cthx+c15.∫1ch2xdx=thx+c16∫dxax+x2=1aarctanxa+c,(a≠0)17.∫dxax−x2=12aln|a+xa−x|+c18.∫xdxax±x2=±12ln|a2±x2|+c19.∫dx√a2−x2=arcsinxa+c,(a0)20.∫dx√x2±a2=ln|x+√x2±a2|+c21.∫xdx√a2±x2=±√a2±x2+c22.∫√a2−x2dx=x2√a2−x2+arcsinxa+c,(a0)22.∫√x2±a2dx=x2√x2±a2±a22ln|x+√x2±a2|+c7.三角学公式sin2θcos2θ1.基本关系1.sinθ∙cscθ=12.cosθ∙secθ=13.tanθ∙cotθ=14.sin2θ+cos2θ=15.sec2θ−tan2θ=16.csc2θ−cot2θ=17.tanθ=nco8.cotθ=con2.两角和与差的三角函数公式1.sin(α±β)=sinαcosβ±cosαsinβ2.cos(α±β)=cosαcosβ∓sinαsinβ3.tan(α±β)=tanα±tanβ1∓tanαtanβ4.cot(α±β)=cotαcotβ∓1cotβ±cotα3.倍角公式1.sinα=sinαcosα=2tanα1+tan2α2.cosα=cos2α−sin2α=cos2α−1=1−sin2α=1−tan2α1+tan2α3.tanα=2tanα1−tan2α4.cotα=cot2α−12cotα5.sin3α=3sinα−4sinα6.cos3α=4cosα−3cosα4.半角公式1.sin2α2=1−coα22.cos2α2=1+coα23.tan2α2=1−coα1+coα=(1−coαnα)2=(nα1+coα)24.cot2α2=1+coα1−coα=(1+coαnα)2=(nα1−coα)25.和差化积公式1.sinα+sinβ=sinα+β2cosα−β22.sinα−sinβ=cosα+β2sinα−β23.cosα+cosβ=cosα+β2cosα−β24.cosα−cosβ=−sinα+β2sinα−β25.tanα±tanβ=±n（α±β）nαnβ6.cotα±cotβ=±co（α∓β）nαnβ7.tanα±cotβ=±co（α∓β）coαnβ6.积化和差公式1.sinαsinβ=−12[cos(α+β)−cos(α−β)]2.cosαcosβ=12[cos(α+β)+cos(α−β)]3.sinαcosβ=12[sin(α+β)+sin(α−β)]7.双曲函数的基本关系1.cosh2t−sinh2t=12.1−tanh2t=1coh2t3.coth2t=1+1nh2t4.sinhx=sinhxcoshx=cosh2x∙tanhx5.sinhx=ex−e−x26.coshx=ex+e−x27.双曲余弦的反函数x=ln(y±√y2−1)8.万能公式1.sinx=2tanx21+tan2x2.cosx=1−tan2x1+tan2x3.tanx=2tanx21−tan2x4.cotx=1−tan2x2tanx25.secx=1+tan2x1−tan2x6.cscx=1+tan2x2tanx2