其他|常用泰勒、微积分公式其他|高数

常用公式常用穷小替换 x = > sin ? x = > tan ? x = > arcsin ? x = > arctan ? x = > ln ? ( x + 1 ) = > e x ? 1 x=> \sin x=> \tan x=> \arcsin x=> \arctan x=> \ln (x+1)=> e^x-1 x=>sinx=>tanx=>arcsinx=>arctanx=>ln(x+1)=>ex?1
( x + 1 ) a ? 1 = > a x (x+1)^a-1=> ax (x+1)a?1=>ax
a x ? 1 = > x l n ( a ) a^x-1=> xln(a) ax?1=>xln(a)
1 ? cos ? x = > 1 2 x 2 1-\cos x=> \frac{1}{2}x^2 1?cosx=>21?x2
tan ? x ? sin ? x = > tan ? x ( 1 ? cos ? x ) = > 1 2 x 3 \tan x-\sin x=> \tan x(1-\cos x)=> \frac{1}{2}x^3 tanx?sinx=>tanx(1?cosx)=>21?x3
常用泰勒展开式

x ? f ( x ) x-f(x) x?f(x)展开
x ? sin ? x = 1 6 x 3 + o ( x 3 ) x-\sin x=\frac{1}{6}x^3+o(x^3) x?sinx=61?x3+o(x3)
x ? arcsin ? x = ? 1 6 x 3 + o ( x 3 ) x-\arcsin x=-\frac{1}{6}x^3+o(x^3) x?arcsinx=?61?x3+o(x3)
x ? tan ? x = ? 1 3 x 3 + o ( x 3 ) x-\tan x=-\frac{1}{3}x^3+o(x^3) x?tanx=?31?x3+o(x3)
x ? arctan ? x = 1 3 x 3 + o ( x 3 ) x-\arctan x=\frac{1}{3}x^3+o(x^3) x?arctanx=31?x3+o(x3)
三角函数展开
e x = 1 + x + x 2 2 ! + x 3 3 ! + o ( x 3 ) e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+o(x^3) ex=1+x+2!x2?+3!x3?+o(x3)
sin ? x = x ? x 3 3 ! + x 5 5 ! + o ( x 5 ) \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+o(x^5) sinx=x?3!x3?+5!x5?+o(x5)
cos ? x = 1 ? x 2 2 ! + x 4 4 ! + o ( x 4 ) \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+o(x^4) cosx=1?2!x2?+4!x4?+o(x4)
ln ? ( x + 1 ) = x ? 1 2 x 2 + 1 3 x 3 + o ( x 3 ) \ln(x+1)=x-\frac{1}{2}x^2+\frac{1}{3}x^3+o(x^3) ln(x+1)=x?21?x2+31?x3+o(x3)

常用微分公式 d tan ? x = ( sec ? x ) 2 d x d\tan x=(\sec x)^2dx dtanx=(secx)2dx
d cot ? x = ? ( csc ? x ) 2 d x d\cot x=-(\csc x)^2dx dcotx=?(cscx)2dx
d sec ? x = sec ? x tan ? x d x d\sec x=\sec x\tan xdx dsecx=secxtanxdx
d csc ? x = ? csc ? x cot ? x d x d\csc x=-\csc x\cot xdx dcscx=?cscxcotxdx
d arcsin ? x = 1 1 ? x 2 d x d\arcsin x=\frac{1}{\sqrt{1-x^2}}dx darcsinx=1?x2 ?1?dx
d arccos ? x = ? 1 1 ? x 2 d x d\arccos x=-\frac{1}{\sqrt{1-x^2}}dx darccosx=?1?x2 ?1?dx
d arctan ? x = 1 1 + x 2 d x d\arctan x=\frac{1}{1+x^2}dx darctanx=1+x21?dx
d a r c o t x = ? 1 1 + x 2 d x darcot x=-\frac{1}{1+x^2}dx darcotx=?1+x21?dx
常用高阶导数公式 ( e a x ) ( n ) = a n e a n (e^{ax})^{(n)}=a^ne^{an} (eax)(n)=anean
( sin ? a x ) ( n ) = a n sin ? ( a x + n Π 2 ) (\sin ax)^{(n)}=a^n\sin (ax+n\frac{\Pi}{2}) (sinax)(n)=ansin(ax+n2Π?)
( cos ? a x ) ( n ) = a n cos ? ( a x + n Π 2 ) (\cos ax)^{(n)}=a^n\cos (ax+n\frac{\Pi}{2}) (cosax)(n)=ancos(ax+n2Π?)
( ln ? ( 1 + x ) ) ( n ) = ( ? 1 ) n ? 1 ( n ? 1 ) ! ( x + 1 ) n (\ln (1+x))^{(n)}=(-1)^{n-1}\frac{(n-1)!}{(x+1)^n} (ln(1+x))(n)=(?1)n?1(x+1)n(n?1)!?
( 1 x ) ( n ) = ( ? 1 ) n n ! x n + 1 (\frac{1}{x})^{(n)}=(-1)^n\frac{n!}{x^{n+1}} (x1?)(n)=(?1)nxn+1n!?

莱布尼茨公式
( u v ) ( n ) = u ( n ) v + C n 1 u ( n ? 1 ) v + C n k u ( n ? k ) v ( k ) + u v n (uv)^{(n)}=u^{(n)}v+C_n^1u^{(n-1)}v+C_n^ku^{(n-k)}v^{(k)}+uv^{n} (uv)(n)=u(n)v+Cn1?u(n?1)v+Cnk?u(n?k)v(k)+uvn

常用积分公式 【其他|常用泰勒、微积分公式】 ∫ tan ? x d x = ? ln ? ∣ cos ? x ∣ + C \int \tan xdx=-\ln|\cos x|+C ∫tanxdx=?ln∣cosx∣+C
∫ cot ? x d x = ln ? ∣ sin ? x ∣ + C \int \cot xdx=\ln|\sin x|+C ∫cotxdx=ln∣sinx∣+C
∫ sec ? x d x = ln ? ∣ sec ? x + tan ? x ∣ + C \int \sec xdx=\ln\left|\sec x+\tan x\right|+C ∫secxdx=ln∣secx+tanx∣+C
∫ csc ? x d x = ln ? ∣ csc ? x ? cot ? x ∣ + C \int \csc x dx=\ln\left|\csc x-\cot x\right|+C ∫cscxdx=ln∣cscx?cotx∣+C
∫ sec ? 2 ( x ) d x = tan ? x + C \int \sec^2(x)dx=\tan x+C ∫sec2(x)dx=tanx+C
∫ csc ? x d x = cot ? x + C \int \csc xdx=\cot x+C ∫cscxdx=cotx+C
∫ 1 a 2 + x 2 d x = 1 a tan ? ( 1 a x ) + C \int \frac{1}{a^2+x^2}dx=\frac{1}{a}\tan(\frac{1}{a}x)+C ∫a2+x21?dx=a1?tan(a1?x)+C
∫ 1 a 2 ? x 2 d x = 1 2 a ln ? ∣ a + x a ? x ∣ + C \int \frac{1}{a^2-x^2}dx=\frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right|+C ∫a2?x21?dx=2a1?ln∣∣?a?xa+x?∣∣?+C
∫ 1 a 2 ? x 2 d x = arcsin ? 1 a x \int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin\frac{1}{a}x ∫a2?x2 ?1?dx=arcsina1?x
∫ 1 x 2 ± a 2 d x = ln ? ∣ x + x 2 ± a 2 ∣ + C \int \frac{1}{\sqrt{x^2\pm a^2}}dx=\ln|x+\sqrt{x^2\pm a^2}|+C ∫x2±a2 ?1?dx=ln∣x+x2±a2 ?∣+C
∫ ln ? x d x = x ln ? x ? x + C \int \ln xdx=x\ln x-x+C ∫lnxdx=xlnx?x+C
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