사용자:Namsoo/연습장/수학공식

logarithm properties

${\text{if}}\ y={\text{log}}_{b}\ x\ {\text{then}}\ b^{y}=x$

${\text{log}}_{b}\ b=1\ {\text{and}}\ {\text{log}}_{b}\ 1=0$

${\text{log}}_{b}\ b^{x}=x$

$b^{{\text{log}}_{b}^{\ }x}=x$

${\text{log}}_{a}x={\frac {{\text{log}}_{b}x}{{\text{log}}_{b}a}}$

${\text{log}}_{b}\left(x^{r}\right)=r\ {\text{log}}_{b}x$

${\text{log}}_{b}(xy)={\text{log}}_{b}x+{\text{log}}_{b}y$

${\text{log}}_{b}\left({\frac {x}{y}}\right)={\text{log}}_{b}x-{\text{log}}_{b}y$

Common Factoring Examples

$x^{2}-a^{2}=(x+a)(x-a)$

$x^{2}+2ax+a^{2}=(x+a)^{2}$

$x^{2}-2ax-a^{2}=(x-a)^{2}$

$x^{2}+(a+b)x+ab=(x+a)(x+b)$

$x^{2}+3ax^{2}+3a^{2}x+a^{3}=(x+a)^{3}$

$x^{3}+a^{3}=(x+a)(x^{2}-ax_{a}^{2})$

$x^{3}-a^{3}=(x-a)(x^{2}+ax+a^{2})$

$x^{2n}-a^{2n}=(x^{n}-a^{n})(x^{n}+a^{n})$

Abbolute Value

$\left|a\right|={\begin{cases}a,{\mbox{if}}\ a\geq 0\\-a,{\mbox{if}}\ a<0\end{cases}}$

$\left|a\right|=\left|-a\right|$

$\left|a\right|\geq 0$

$\left|ab\right|=\left|a\right|\left|b\right|$

$\left|{\frac {a}{b}}\right|={\frac {\left|a\right|}{\left|b\right|}}$

$\left|a+b\right|\leq \left|a\right|+\left|b\right|$

Calculus (integrals)

Definite integral definition

$\int _{a}^{b}f(x)dx=\lim _{n\to \infty }\sum _{k=1}^{n}f(x_{k})\Delta x$

${\mbox{where}}\ \Delta x={\frac {b-a}{n}}\ {\mbox{and}}\ x_{k}=a+k\Delta x$

Fundamental Theorem of Calculus

$\int _{a}^{b}f(x)dx=[F(x)]_{a}^{b}=F(b)-F(a)$

Integration Properties

$\int _{a}^{b}cf(x)dx=c\int _{a}^{b}f(x)dx$

$\int _{a}^{b}f(x)\pm g(x)dx=\int _{a}^{b}f(x)dx\pm \int _{a}^{b}g(x)dx$

$\int _{a}^{b}f(x)dx=0\ {\mbox{and}}\ \int _{a}^{b}f(x)dx=-\int _{b}^{a}f(x)dx$

$\int _{a}^{b}f(x)dx+\int _{b}^{c}f(x)dx=\int _{a}^{c}f(x)dx$

common integrals

$\int k\ dx=kx+C$

$\int x^{n}dx={\frac {1}{n+1}}x^{n+1}+C,n\neq -1$

$\int x^{-1}dx=\int {\frac {1}{x}}dx={\mbox{ln}}\left|x\right|+C$

$\int {\frac {1}{ax+b}}dx={\frac {1}{a}}{\mbox{ln}}\left|ax+b\right|+C$

$\int {\mbox{ln}}(x)\ dx=x\ {\mbox{ln}}(x)-x+C$

$\int e^{x}\ dx=e^{x}+C$

$\int \cos x\ dx=\sin x+C$

$\int \sin x\ dx=-\cos x+C$

$\int \sec ^{2}x\ dx=\tan x+C$

$\int \sec x\tan x\ dx=\sec x+C$

$\int \csc x\cot x\ dx=-\csc x+C$

$\int \csc ^{2}x\ dx=-\cot x+C$

$\int \tan x\ dx=\ln \left|\sec x\right|+C$

$\int \sec x\ dx=\ln \left|\sec x+\tan x\right|+C$

$\int {\frac {1}{a^{2}+u^{2}}}dx={\frac {1}{a}}\tan ^{-1}\left({\frac {u}{a}}\right)+C$

$\int {\frac {1}{\sqrt {a^{2}-u^{2}}}}dx=\sin ^{-1}\left({\frac {u}{a}}\right)+C$

Approximating Definite Integrals

Left-hand and right-hand rectangle approximations

$L_{n}=\Delta x\sum _{k=0}^{n-1}f(x_{k})\qquad \qquad R_{n}=\Delta x\sum _{k=1}^{n}f(x_{k})$

Midpoint Rule

$M_{n}=\Delta x\sum _{k=0}^{n-1}f({\frac {x_{k}+k_{k+1}}{2}})$

Trapezoid Rule

$T_{n}={\frac {\Delta x}{2}}(f(x_{0})+2f(x_{1})+2f(x_{2})+\cdots +f(x_{n}))$

Trignometric substitution

EXPRESSION	SUBSTITUTION	EXPRESSION EVALUATION	IDENTITY USED
${\sqrt {a^{2}-x^{2}}}$	$x=a\sin \theta$ $dx=a\cos \theta \ d\theta$	${\sqrt {a^{2}-a^{2}\ \sin ^{2}\ \theta }}=a\cos \theta$ ${\displaystyle }$	$1-\sin ^{2}\ \theta =\cos ^{2}\ \theta$ ${\displaystyle }$
${\sqrt {x^{2}-a^{2}}}$	$x=a\sec \theta$ $dx=a\sec \theta \tan \theta \ d\theta$	${\sqrt {a^{2}\sec ^{2}\theta -a^{2}}}=a\tan \theta$ ${\displaystyle }$	$\sec ^{2}\theta -1=\tan ^{2}\theta$ ${\displaystyle }$
${\sqrt {a^{2}+x^{2}}}$	$x=a\tan \theta$ $dx=a\sec ^{2}\theta \ d\theta$	${\sqrt {a^{2}+a^{2}\tan ^{2}\theta }}=a\ \sec ^{2}\theta$ ${\displaystyle }$	$1+\tan ^{2}\theta =\sec ^{2}\theta$ ${\displaystyle }$

approximation by simpson rule for even n

$S_{n}={\frac {\Delta x}{3}}(f(x_{0})+4f(x_{1})+2f(x_{2})+\cdots +2f(x_{n-2})+4f(x_{n-1}+f(x_{n}))$

integration by substitution

$\int _{a}^{b}f(g(x))g'(x)dx=\int _{g(a)}^{g(b)}f(u)du$

Where $u=g(x)$ and $du=g'(x)dx$

integration by parts

$\int u\ dv=uv-\int v\ du\ {\mbox{where}}\ v=\int dv$

or

$\int f(x)g'(x)dx=f(x)g(x)-\int f'(x)g(x)dx$

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${\frac {d}{dx}}\left(f(x)\right)=f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

$(cf(x))'=c(f'(x))$

$(f(x)\pm g(x))'=f'(x)\pm g'(x)$

${\frac {d}{dx}}\left(c\right)=0$

If $f$ is differentiable on the interval ($a,b$) and continuous at the end points lhere exists a $c$ in ($a,b$) such that

$f'\left(c\right)={\frac {f(b)-f(a)}{b-a}}$

$(f(x)g(x))'=f(x)'g(x)+f(x)g(x)'$

${\frac {d}{dx}}\left({\frac {f(x)}{d(x)}}\right)={\frac {f'(x)g(x)-f(x)g'(x)}{[g(x)]^{2}}}$