1、常用數學與微積分公式定理( 1/ 7)常用數學公式常用數學公式axdttxspecial casesbxyxy xrxxyxyxr)ln ( )ln( ) , ln( ) , ln( ) ln( ) ln ln ln( ) lnln( ) ln ln=+=+ =z1010 0:ae e e jbe xce y y yey y yjxyy). , ,cossin)exp()xp(ln)ln ln (exp( )ln102 718281828 1=+= = =sin( ) sin cos cos sinsin( ) sin cos cos sincos( ) cos cos sin sincos(
2、 ) cos cos sin sinsin cos sin( ) sin( )cos sin sin( ) sin( )cos cos cos( ) cos( )sin sin cos( ) cos( )()/(AB A B A BAB A B A BAB A B A BAB A B A BA B AB ABAB AB ABAB AB ABAB AB ABassumexAByABthenAxyBx+= + = += = + = += += + = + =+=RST=+=121212122+RST+=+=+=+=+yxyxy xyxyxy xyxyxy xyxyxy xy)/sin sin si
3、n cossin sin cos sincos cos cos coscos cos sin sin2222222222222sin sin coscos cos sin cos sincoscos,sincossin cossin tan sec , cos cot csc221212212211122 2 222 222 222 = =+=+=+= +=“aaabgb gch b gch常用數學與微積分公式定理( 2/ 7)常用微分公式常用微分公式d f g g df f dgdu du d u udCdxdC C constantdx C dx dx dx C()()(: )() ()=
4、+=+= =+= = +12 1200dedxedeedxdxdx x udu d uxxx= = =lnln11dxdxnx dx nx dxdxdxxdxxdddx x xdxxdxnnnn= =FHGIKJ=FHGIKJ=1122222211 11dxy ydx xdydx y mx ydx ny xdym ydx n xdydxyxydyxxdy ydxxdxyydx xdyymn m n n mmnmn()()=+= + + =FHGIKJ=FHGIKJ=ch111122dxdxxdxxdxdxdxxdxxdxdxdxxdxxdxdxdxxdx xdxdxdxxx dx xxdxdx
5、dxxx dx xxdxsincos sin coscossin cos sintansec tan seccotcsc cot cscsecsec tan sec sec tancsccsc cot csc csc cot= = = = =22dxdxxdxxdxdxdx xdxxdxdxdxxxdxxxdxdxdxxdxxdxdxdx xdxxdxdxdxxxdxxxdxsinsintantansecseccoscoscotcotcsccsc=+=+=122112211221122112211221111111111111111111常用數學與微積分公式定理(3/7)微積分定理與公式微積分
6、定理與公式ddxfx gxddxfxddxgxddxfg gddxffddxgfgfgfgfgfg fgggxyyu uuxdydxdydududx() () () ()()() ()= = + = +LNMOQP= =20chain rule : if and then dF xdxfx fxdx Fx Cddxf t dt f x addxfxdx fx d fxdx fxdxfxdx d fxdxdf xdxdx f x C d f x f x C Cyyx dyydxuuxy duuxdxuydyudv uv vduax()() () ()() ( ) , :() () () ()()
7、 ()()() () () ( : )()(,)= =+= =+= =+=+RS|T|=zzzzzzzzzconstant.integral constant( integral by parts )() ()ddxfxtdt fxbxdbdxfxpxdpdx xfxtdtpxbxpxbx(,) ,() , () (,)() ()zz= +b g b gdCdxdxdxnxnn=01dedxededxaedadxaaxx chain ruleaxaxxx= =ln常用數學與微積分公式定理(4/7)微積分定理與公式微積分定理與公式dxdxxdxdxxdxdxxdxdxxdxdxxdxdxxdxd
8、xxdxdxxdxdxxxdxdxxxdxdxxxdxdxxchain rulechain rulechain rulechain rulechain rulechain rulesincossincoscossincossintansectanseccotcsccotcscsecsec tansecsec tancsccsc cotcsccsc= = = = = = = 22cotxdxdxxdxdxxdxdxxdxdxxdxdx xdxdx xdxdx xdxdx xdxdxxxdxdxxchain rulechain rulechain rulechain rulechain rules
9、in sin()cos cos()tan tan()cot cot()sec sec(= = =+ =+=+ =+= =121212121212121212111111111111xdxdxxxdxdxxxchain rule)csc csc()2121211111= =Binomial formulaxy Cxy Cx yLeibniz s formuladdxfg fg Cf gwhere Cnknknknknknknkknknnk knnnknknknkkn:!()() ( )+= = =FHGIKJ=bgbgbgbg000常用數學與微積分公式定理(5/7)微積分定理與公式* Taylo
10、rs series expansion :fxfanxafafaxafaxafaxawhere n n n nf(x) is an infinitely differentiable functionexnxx xxxnxxnnnxnnnnn()()!()()()!()()!()()!()!.!sin()()!()()=+ += = + + + +=+=+=0233023210121221112 3121“b gb g b gb gsome important expansions:35720246233571212461112123!cos()()! ! ! !+ += + +=+=xxxx
11、nxxxxpxppxpp pxnnnpbgbg bgb g“If thenddxFx f x f xdx Fx CddxCbg bg bg bg=+=LNMOQPz,0xdxnxC ndxxxCduuuCnn=+=+ =+zzz1111bgln lnedxaeCadxaaC ae eax axxx xaxax= +=+= =zz11lnlnlnej111111111111212121212121=+=+=+=+=+=+zzzzxdx x Cxdx x Cxdx x Cxdx x Cxxdx x Cxxdx x Csin costan cotsec csc常用數學與微積分公式定理(6/7)微積分
12、定理與公式ebxdxeababxbbxCebxdxeababxbbxCaxaxaxaxcos cos sinsin sin cos=+ +=+ +zz2222sin coscos sintan ln sec sec tancot ln sin csc cotsec ln sec tan sec tancsc ln csc cot csc cotsin coscos sintan ln seccot ln sinxdx x Cxdx x Cxdx x C xdx x Cxdx x C xdx x Cxdx x x C xdx x Cxdx x x C xdx x Cxdx x Cxdx x Cxdx x Cxdxzzzzzzzzzz= +=+=+ =+=+ =+=+ = += + + = += += +=+=2222111111xCxdx x x Cxdx x x C+=+= + +zzsec ln sec tancsc ln csc cot11
