常用微积分公式

2023年12月17日发(作者：文山高三数学试卷)

常用數學與微積分公式定理a)lnx=dt(x>0)1tspecial cases:ln(1)=0,ln(0)=−∞,ln(∞)=+∞x(1/7)z常用數學公式b)ln(xy)=lnx+lnyxln()=lnx−lnyyln(xr)=r⋅lnxa)e1≈2.718281828,e0=1,ejθ=cosθ+jsinθb)ex=exp(x)c)elny=y⇔exp(lny)=ylney=y⇔ln(exp(y))=ysin(A+B)=sinA⋅cosB+cosA⋅sinBsin(A−B)=sinA⋅cosB−cosA⋅sinBcos(A+B)=cosA⋅cosB−sinA⋅sinBcos(A−B)=cosA⋅cosB+sinA⋅sinB1sinA⋅cosB=sin(A+B)+sin(A−B)21cosA⋅sinB=sin(A+B)−sin(A−B)21cosA⋅cosB=cos(A+B)+cos(A−B)21sinA⋅sinB=−cos(A+B)−cos(A−B)2x=A+BA=(x+y)/2assumetheny=A−BB=(x+y)/2RSTRSTx+yx−y⋅cos22x+yx−ysinx−siny=2⋅cos⋅sin22x+yx−ycosx+cosy=2⋅cos⋅cos22x+yx−ycosx−cosy=−2⋅sin⋅sin22sinx+siny=2⋅sinsin2θ=2sinθcosθcos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ1+cos2θ1−cos2θcos2θ=,sin2θ=⇔sin2θ+cos2θ=1\"a22a÷sin2θ⇒1+tan2θ=sec2θ,a÷cos2θ⇒1+cot2θ=csc2θbgcbghcbgh常用數學公式常用數學與微積分公式定理常用微分公式d(fg)=gdf+fdgdu1+du2=d(u1+u2)dC=0⇔dC=0(C:constant)dxd(x+C)=dx⇔dx=d(x+C)(2/7)dex=ex⇔dex=exdxdx1dlnx1=⇔du=dlnudxxud(xy)=ydx+xdydx=nxn−1⇔dxn=nxn−1dxdxdx2=2x⇔dx2=2xdxdxd1−11−1=2⇔d=2dxdxxxxxndxmyn=m⋅xm−1yndx+n⋅yn−1xmdyd(xmyn)∴m⋅ydx+n⋅xdy=m−1n−1xychFGIJHKFGIJHKFGyIJ=xdy−ydxHxKxFxIydx−xdydGJ=HyKyd22dsinx=cosx⇔dsinx=cosxdxdxdcosx=−sinx⇔dcosx=−sinxdxdxdtanx=sec2x⇔dtanx=sec2xdxdxdcotx=−csc2x⇔dcotx=−csc2xdxdxdsecx=secxtanx⇔dsecx=secxtanxdxdxdcscx=−cscxcotx⇔dcscx=−cscxcotxdxdxdsin−1x1dx=⇔=dsin−1xdx1−x21−x21dtan−1xdx=⇔=dtan−1x221+x1+xdx1dsec−1xdx=⇔=dsec−1xdxxx2−1xx2−1−1−dxdcos−1x=⇔=dcos−1xdx1−x21−x2−1−dxdcot−1x=⇔=dcot−1x221+x1+xdx−1−dxdcsc−1x=⇔=dcsc−1xdxxx2−1xx2−1常用微分公式常用數學與微積分公式定理微積分定理與公式(3/7)LMOPNQdddf(x)±g(x)=f(x)±g(x)dxdxdxdddf⋅g=g⋅f+f⋅g⇔f⋅g′=f′⋅g+f⋅g′dxdxdxf′f′⋅g−f⋅g′=⇒g(x)≠0gg2dydydu=⋅dxdudxchain rule:if y=y(u)andu=u(x)then

dF(x)f(x)dx=F(x)+C=f(x)⇒dxdxf(t)dt=f(x),a:(x)dx=f(x)=>df(x)dx=f(x)dxdxzzzf(x)dx=dR|S|Tzzzzf(x)dxdf(x)dx=f(x)+C=>df(x)=f(x)+Cdxy=y(x)dy=y′⋅dx∂u∂uu=u(x,y)du=⋅dx+⋅dy∂x∂yu⋅dv=u⋅v−v⋅duz(C:integral constant)ddxzz( integralby parts)b(x)p(x)f(x,t)dt=fx,b(x)bgdbdp−fx,p(x)+dxdxbgz∂f(x,t)dtp(x)∂xb(x)dC=0dxdxn=nxn−1dxdexrule=ex⎯chain⎯⎯⎯→dxdax=lna⋅axdxdeax=aeaxdx微積分定理與公式常用數學與微積分公式定理微積分定理與公式(4/7)dsinxdsinωxrule=cosx⎯chain⎯⎯⎯→=ω⋅cosωxdxdxdcosxdcosωxrule=−sinx⎯chain⎯⎯⎯→=−ω⋅sinωxdxdxdtanxdtanωxrule=sec2x⎯chain⎯⎯⎯→=ω⋅sec2ωxdxdxdcotxdcotωxrule=−csc2x⎯chain⎯⎯⎯→=−ω⋅csc2ωxdxdxdsecxdsecωxrule=secx⋅tanx⎯chain⎯⎯⎯→=ω⋅secωx⋅tanωxdxdxdcscxdcscωxrule=−cscx⋅cotx⎯chain⎯⎯⎯→=−ω⋅cscωx⋅cotωxdxdxdsin−1x1=dx1−x2−1dcos−1x=dx1−x2dtan−1x1=dx1+x2−1dcot−1x=dx1+x2⎯⎯⎯⎯→⎯⎯⎯⎯→⎯⎯⎯⎯→⎯⎯⎯⎯→chainrulechainrulechainrulechainrulechainruledsin−1ωxω=dx1−(ωx)2−ωdcos−1ωx=dx1−(ωx)2dtan−1ωxω=dx1+(ωx)2−ωdcot−1ωx=dx1+(ωx)2dsec−1ωx1=dxx(ωx)2−1dcsc−1ωx−1=dxx(ωx)2−1dsec−1x1=dxxx2−1dcsc−1x−1=dxxx2−1Binomialformula:⎯⎯⎯⎯→⎯⎯⎯⎯→chainrulebx+ygn=∑Cxynkkk=0nn−k=∑Cknxn−kykk=0nnLeibniz\'sformula:bgbg=∑CfFnIn!whereC=GJ=HkKk!bn−kg!dnf⋅g=f⋅gndxnk(n)nkk=0(k)g(n−k)微積分定理與公式常用數學與微積分公式定理*Taylor’sseries expansion:f(x)=∑n=0∞(5/7)f(n)(a)⋅(x−a)nn!f′(a)f′′(a)f(3)(a)2=f(a)+(x−a)+(x−a)+(x−a)3+\"\"1!2!3!wheren!=nn−1n−2\"21.f(x)portantexpansions:xnxx2x3e=∑=1++++⋅⋅⋅⋅⋅n!1!2!3!n=0x∞bgbgbgbg(−1)nx2n+1x3x5x7sinx=∑=x−+−+⋅⋅⋅⋅⋅n(+)!213!5!7!n=0∞x2x4x6(−1)nx2ncosx=∑=1−+−+⋅⋅⋅⋅⋅(2n)!2!4!6!n=0∞b1+xgIfp=1+px+pp−12pp−1p−23x+x+\"\"2!3!bgbgbbggdFx=fx,thendxxndx=1n+1x+Cn+1bgbgzfxdx=Fx+Cbg⇒LMdC=0OPNdxQzzzzzzzbn≠1gdx=lnx+C⇒xeaxdx=zdu=lnu+Cu1ax⋅e+Ca1lneaxjxxxadx=⋅a+C⇒a=e=ex⋅lnalna11−x21−1dx=tanx+C21+x1dx=sec−1x+Cxx2−1dx=sin−1x+Czzz−11−x2−1−1dx=cotx+C21+x−1dx=csc−1x+Cxx2−1dx=cos−1x+C微積分定理與公式常用數學與微積分公式定理(6/7)zzzzzzzzzzzzzzeaxecosbxdx=2a⋅cosbx+b⋅sinbx+Ca+b2eaxaxesinbxdx=2a⋅sinbx−b⋅cosbx+Ca+b2axsinxdx=−cosx+Ccosxdx=sinx+Ctanxdx=lnsecx+C⇒cotxdx=lnsinx+C⇒zzsec2xdx=tanx+Ccsc2xdx=−cotx+Csecxdx=lnsecx+tanx+C⇒cscxdx=−lncscx+cotx+C⇒sinωxdx=−cosωxdx=tanωxdx=cotωxdx=secωxdx=1zsec2ωxdx=1zωtanωx+C1csc2ωxdx=−ωcotωx+Cω⋅cosωx+C1ω1⋅sinωx+Clnsecωx+Clnsinωx+Clnsecωx+tanωx+C1ω1ω1ωcscωxdx=−ωlncscωx+cotωx+C微積分定理與公式