Fórmulas de física en lenguaje Latex 13 - Matemática, Derivadas, Integrales y Transformadas de Laplace

[Richard R Richard]

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Esta entrega esta dedicada a la Matemática, Derivadas, integrales y Transformadas de Laplace

Derivadas
$\dfrac {\mathrm{d} C}{\mathrm{d} x}=0$	\dfrac {\mathrm{d} C}{\mathrm{d} x}=0
$\dfrac {\mathrm{d} x}{\mathrm{d} x}=1$	\dfrac {\mathrm{d} x}{\mathrm{d} x}=1
$\dfrac {\mathrm{d} \left ( x^n \right )}{\mathrm{d} x}=nx^{n-1}$	\dfrac {\mathrm{d} \left ( x^n \right )} {\mathrm{d} x}=nx^{n-1}
$\dfrac {\mathrm{d} \left ( ln (x) \right )}{\mathrm{d} x}=x^{-1}$	\dfrac {\mathrm{d} \left ( ln (x) \right )} {\mathrm{d} x}=x^{-1}
$\dfrac {\mathrm{d} \left ( a^x \right )}{\mathrm{d} x}=a^x ln (a)$	\dfrac {\mathrm{d} \left ( a^x \right )} {\mathrm{d} x}=a^x ln (a)
$\dfrac {\mathrm{d} \left ( e^x \right )}{\mathrm{d} x}=e^x$	\dfrac {\mathrm{d} \left ( e^x \right )} {\mathrm{d} x}=e^x
$\dfrac {\mathrm{d} \left ( \sin ax \right )}{\mathrm{d} x}=a\cos ax$	\dfrac {\mathrm{d} \left ( \sin ax \right )} {\mathrm{d} x}=a\cos ax
$\dfrac {\mathrm{d} \left ( \cos ax \right )}{\mathrm{d} x}=-a\sin ax$	\dfrac {\mathrm{d} \left ( \cos ax \right )} {\mathrm{d} x}=-a\sin ax
$\dfrac {\mathrm{d} \left ( \tan ax \right )}{\mathrm{d} x}=\dfrac a{\cos^2 ax}=a\sec^2ax$	\dfrac {\mathrm{d} \left ( \tan ax \right )} {\mathrm{d} x}=\dfrac a{\cos^2 ax}=a\sec^2ax
$\dfrac {\mathrm{d} \left ( \cot x \right )}{\mathrm{d} x}=-\dfrac 1{\sin^2 x}=-\csc^2x$	\dfrac {\mathrm{d} \left ( \cot x \right )} {\mathrm{d} x}=-\dfrac 1{\sin^2 x}=-\csc^2x
$\dfrac {\mathrm{d} \left ( \sec x \right )}{\mathrm{d} x}=\sec x \tan x$	\dfrac {\mathrm{d} \left ( \sec x \right )} {\mathrm{d} x}=\sec x \tan x
$\dfrac {\mathrm{d} \left ( \csc x \right )}{\mathrm{d} x}=-\csc x \cot x$	\dfrac {\mathrm{d} \left ( \csc x \right )} {\mathrm{d} x}=-\csc x \cot x
$\dfrac {\mathrm{d} \left ( \arcsin x \right )}{\mathrm{d} x}=\dfrac 1{\sqrt{1-x^2}}$	\dfrac {\mathrm{d} \left ( \arcsin x \right )} {\mathrm{d} x}=\dfrac 1{\sqrt{1-x^2}}
$\dfrac {\mathrm{d} \left ( \arccos x \right )}{\mathrm{d} x}=\dfrac {-1}{\sqrt{1-x^2}}$	\dfrac {\mathrm{d} \left ( \arccos x \right )} {\mathrm{d} x}=\dfrac {-1}{\sqrt{1-x^2}}
$\dfrac {\mathrm{d} \left ( \arctan x \right )}{\mathrm{d} x}=\dfrac 1{1+x^2}$	\dfrac {\mathrm{d} \left ( \arctan x \right )} {\mathrm{d} x}=\dfrac 1{1+x^2}
$\dfrac {\mathrm{d} \left ( arccot\:x \right )}{\mathrm{d} x}=\dfrac {-1}{1+x^2}$	\dfrac {\mathrm{d} \left ( arccot\:x \right )} {\mathrm{d} x}=\dfrac {-1}{1+x^2}
$\dfrac {\mathrm{d} \left ( arcsec\:x \right )}{\mathrm{d} x}=\dfrac 1{x\sqrt{x^2-1}}$	\dfrac {\mathrm{d} \left ( arcsec\:x \right )} {\mathrm{d} x}=\dfrac 1{x\sqrt{x^2-1}}
$\dfrac {\mathrm{d} \left ( arccsc\:x \right )}{\mathrm{d} x}=\dfrac {-1}{x\sqrt{x^2-1}}$	\dfrac {\mathrm{d} \left ( arccsc\:x \right )} {\mathrm{d} x}=\dfrac {-1}{x\sqrt{x^2-1}}
$\dfrac {\mathrm{d} \left ( \sinh x \right )}{\mathrm{d} x}=\cosh x$	\dfrac {\mathrm{d} \left ( \sinh x \right )} {\mathrm{d} x}=\cosh x
$\dfrac {\mathrm{d} \left ( \cosh x \right )}{\mathrm{d} x}=\sinh x$	\dfrac {\mathrm{d} \left ( \cosh x \right )} {\mathrm{d} x}=\sinh x
$\dfrac {\mathrm{d} \left ( \tanh x \right )}{\mathrm{d} x}=\dfrac 1{\cosh^2 x}=sech^2x$	\dfrac {\mathrm{d} \left ( \tanh x \right )} {\mathrm{d} x}=\dfrac 1{\cosh^2 x}=sech^2x
$\dfrac {\mathrm{d} \left ( \coth x \right )}{\mathrm{d} x}=-\dfrac 1{\sinh^2 x}=csch^2x$	\dfrac {\mathrm{d} \left ( \coth x \right )} {\mathrm{d} x}=-\dfrac 1{\sinh^2 x}=csch^2x
$\dfrac {\mathrm{d} \left ( sech\:x \right )}{\mathrm{d} x}=-sech\:x \tanh x$	\dfrac {\mathrm{d} \left ( sech x \right )} {\mathrm{d} x}=-sech\:x \tanh x
$\dfrac {\mathrm{d} \left ( csch\:x \right )}{\mathrm{d} x}=-csch\:x \coth x$	\dfrac {\mathrm{d} \left ( csch x \right )} {\mathrm{d} x}=-csch\:x \coth x
$\dfrac {\mathrm{d} \left ( arcsinh\:x \right )}{\mathrm{d} x}=\dfrac 1{\sqrt{1+x^2}}$	\dfrac {\mathrm{d} \left ( arcsinh\:x \right )} {\mathrm{d} x}=\dfrac 1{\sqrt{1+x^2}}
$\dfrac {\mathrm{d} \left ( arccosh\:x \right )}{\mathrm{d} x}=\dfrac {-1}{\pm \sqrt{x^2-1}}$	\dfrac {\mathrm{d} \left ( arccosh\:x \right )} {\mathrm{d} x}=\dfrac {-1}{\pm \sqrt{x^2-1}}
$\dfrac {\mathrm{d} \left ( arctanh\:x \right )}{\mathrm{d} x}=\dfrac 1{1-x^2}$	\dfrac {\mathrm{d} \left ( arctanh\:x \right )} {\mathrm{d} x}=\dfrac 1{1-x^2}

Propiedades de derivadas
Multiplicacion por escalar	$\dfrac {\mathrm{d} (Cy)}{\mathrm{d} x}=C \dfrac {\mathrm{d} y}{\mathrm{d} x}$	\dfrac {\mathrm{d} (Cy)}{\mathrm{d} x}= C \dfrac {\mathrm{d} y}{\mathrm{d} x}
Suma y distribución	$\dfrac {\mathrm{d}( y+z)}{\mathrm{d} x}= \dfrac {\mathrm{d} y}{\mathrm{d} x}+\dfrac {\mathrm{d} z}{\mathrm{d} x}$	\dfrac {\mathrm{d} (y+z)}{\mathrm{d} x}= \dfrac {\mathrm{d} y}{\mathrm{d} x}+ \dfrac {\mathrm{d} z}{\mathrm{d} x}
Derivada del producto	$\dfrac {\mathrm{d} (y \cdot z)}{\mathrm{d} x}= z\dfrac {\mathrm{d} y}{\mathrm{d} x}+y\dfrac {\mathrm{d} z}{\mathrm{d} x}$	\dfrac {\mathrm{d} (y \cdot z)}{\mathrm{d} x}= z\dfrac {\mathrm{d} y}{\mathrm{d} x}+ y\dfrac {\mathrm{d} z}{\mathrm{d} x}
Derivada de la división	$\dfrac {\mathrm{d} (\dfrac yz)}{\mathrm{d} x}= \dfrac {z\dfrac {\mathrm{d} y}{\mathrm{d} x}-y\dfrac {\mathrm{d} z}{\mathrm{d} x}}{z^2}$	\dfrac {\mathrm{d} (\dfrac yz)}{\mathrm{d} x}= \dfrac {z\dfrac {\mathrm{d} y}{\mathrm{d} x}- y\dfrac {\mathrm{d} z}{\mathrm{d} x}}{z^2}
Regla de la cadena	$\dfrac {\mathrm{d} (f(g(x)))}{\mathrm{d} x}= f'(g(x))\dfrac {\mathrm{d} g(x)}{\mathrm{d} x}$	\dfrac {\mathrm{d} (f(g(x)))}{\mathrm{d} x}= f'(g(x))\dfrac {\mathrm{d} g(x)}{\mathrm{d} x}

Integrales
$\dst \int a \:\dd x= ax+C$	\dst \int a \:\dd x= ax+C
$\dst \int a f(x) \:\dd x= a \int f(x)\:\dd x$	\dst \int a f(x) \:\dd x= a \int f(x)\:\dd x
$\dst \int (f + g) \:\dd x= \int f \:\dd x+\int g \:\dd x$	\dst \int (f + g) \:\dd x= \int f \:\dd x+\int g \:\dd x
$\dst \int f \dd g= fg- \int g \dd f$	\dst \int f \dd g= fg- \int g \dd f
$\dst \int x^n \:\dd x= \dfrac { x^{n+1}}{n+1}+C$ $n\neq1$	\dst \int x^n \:\dd x= \dfrac { x^{n+1}}{n+1} +C n\neq1
$\dst \int x^{-1} \:\dd x=\ln|x|+C$	\dst \int x^{-1} \:\dd x=\ln|x|+C
$\dst \int e^{x} \:\dd x=e^{x}+C$	\dst \int e^{x} \:\dd x=e^{x}+C
$\dst \int a^{x} \:\dd x=\dfrac {a^{x}}{ln a}+C$	\dst \int a^{x} \:\dd x= \dfrac {a^{x}}{ln a}+C
$\dst \int xa^{x} \:\dd x=\dfrac {a^{x}}{ln a}\cdot \left ( x-\dfrac 1{\ln a}\right ) +C$	\dst \int xa^{x} \:\dd x= \dfrac {a^{x}}{ln a}\cdot \left ( x-\dfrac 1{\ln a}\right ) +C
$\dst \int xe^{x} \:\dd x=e^{x}\cdot \left ( x- 1 \right ) +C$	\dst \int xe^{x} \:\dd x=e^{x}\cdot \left ( x- 1 \right ) +C
$\dst \int \ln x \:\dd x=x\cdot \ln x - x +C=x\cdot (\ln x - 1) +C$	\dst \int \ln x \:\dd x=x\cdot \ln x - x +C =x\cdot (\ln x - 1) +C
$\dst \int x \ln x \:\dd x=\dfrac {x^2}4 \cdot (2 \ln x - 1) +C$	\dst \int x \ln x \:\dd x=\dfrac {x^2}4 \cdot (2 \ln x - 1) +C
$\dst \int \sin x \:\dd x=-\cos x +C$	\dst \int \sin x \:\dd x=-\cos x +C
$\dst \int \cos x \:\dd x=\sin x +C$	\dst \int \cos x \:\dd x=\sin x +C
$\dst \int \sec x \tan x \:\dd x=\sec x +C$	\dst \int \sec x \tan x \:\dd x=\sec x +C
$\dst \int \csc x \cot x \:\dd x=-\csc x +C$	\dst \int \csc x \cot x \:\dd x=-\csc x +C
$\dst \int \tan x \:\dd x=-\ln|\cos x| +C=\ln |\sec x|+C$	\dst \int \tan x \:\dd x=-\ln |\cos x| +C=\ln |\sec x| +C
$\dst \int \cot x \:\dd x=\ln |\cos x| +C=-\ln |\csc x| +C$	\dst \int \cot x \:\dd x=\ln |\cos x| +C=-\ln |\csc x| +C
$\dst \int \sec x \:\dd x=-\ln |\sec x+ \tan x| +C$	\dst \int \sec x \:\dd x=-\ln |\sec x+ \tan x| +C
$\dst \int \csc x \:\dd x=-\ln |\csc x- \cot x| +C$	\dst \int \csc x \:\dd x=-\ln |\csc x- \cot x| +C
$\dst \int \sin^2 x \:\dd x=\dfrac x2-\dfrac 14\sin 2x +C$	\dst \int \sin^2 x \:\dd x=\dfrac x2 -\dfrac 14\sin 2x +C
$\dst \int \cos^2 x \:\dd x=\dfrac x2+\dfrac 14\sin 2x +C$	\dst \int \cos^2 x \:\dd x=\dfrac x2 +\dfrac 14\sin 2x +C
$\dst \int \tan^2 x \:\dd x=\tan x- x +C$	\dst \int \tan^2 x \:\dd x= \tan x- x +C
$\dst \int \cot^2 x \:\dd x=-\cot x- x +C$	\dst \int \cot^2 x \:\dd x= -\cot x- x +C
$\dst \int \sec^2 x \:\dd x=\tan x +C$	\dst \int \sec^2 x \:\dd x= \tan x +C
$\dst \int \csc^2 x \:\dd x=-\cot x +C$	\dst \int \csc^2 x \:\dd x= -\cot x +C
$\dst \int x\sin x \:\dd x=\sin x-x\cos x +C$	\dst \int x\sin x \:\dd x= \sin x-x\cos x +C
$\dst \int x\cos x \:\dd x=\cos x+x\sin x +C$	\dst \int x\cos x \:\dd x= \cos x+x\sin x +C
$\dst \int \arcsin x \:\dd x=x\sin x+\sqrt{1-x^2} +C$	\dst \int \arcsin x \:\dd x=x\sin x+ \sqrt{1-x^2} +C
$\dst \int \arccos x \:\dd x=x\cos x-\sqrt{1-x^2} +C$	\dst \int \arccos x \:\dd x=x\cos x- \sqrt{1-x^2} +C
$\dst \int \arctan x \:\dd x=x\tan x-\ln(\sqrt{1+x^2}) +C$	\dst \int \arctan x \:\dd x=x\tan x- \ln(\sqrt{1+x^2}) +C
$\dst \int arccot\:x \:\dd x=x\cot x+\ln(\sqrt{1+x^2}) +C$	\dst \int arccot\:x \:\dd x=x\cot x+ \ln(\sqrt{1+x^2}) +C
$\dst \int arcsec\:x \:\dd x=x\sec x-\ln(x+\sqrt{x^2-1}) +C=x\sec x-\arccosh x +C$	\dst \int arcsec\:x \:\dd x= x\sec x-\ln(x+\sqrt{x^2-1}) +C= x\sec x-\arccosh x +C
$\dst \int arccsc\:x \:\dd x=x\csc x+\ln(x+\sqrt{x^2-1}) +C=x\sec x+\arccosh x +C$	\dst \int arccsc\:x \:\dd x= x\csc x+\ln(x+\sqrt{x^2-1}) +C= x\sec x+\arccosh x +C
$\dst \int \sinh x \:\dd x=\cosh x +C$	\dst \int \sinh x \:\dd x=\cosh x +C
$\dst \int \cosh x \:\dd x=\sinh x +C$	\dst \int \cosh x \:\dd x=\sinh x +C
$\dst \int sech^2 x \:\dd x=\tanh x +C$	\dst \int sech^2 x \:\dd x= \tanh x +C
$\dst \int csch^2 x \:\dd x=-\coth x +C$	\dst \int csch^2 x \:\dd x= -\coth x +C
$\dst \int sech\:x \tanh x\:\dd x=-sech\:x +C$	\dst \int sech x \tanh x\:\dd x= -sech\:x +C
$\dst \int csch\:x \coth x\:\dd x=-csch\:x +C$	\dst \int csch x \coth x\:\dd x= -csch\:x +C
$\dst \int \tanh x\:\dd x= \ln (\cosh x )+C$	\dst \int \tanh x\:\dd x= \ln (\cosh x )+C
$\dst \int \coth x\:\dd x= \ln |\sinh x |+C$	\dst \int \coth x\:\dd x= \ln |\sinh x |+C
$\dst \int sech\:x \:\dd x= \arctan (\sinh x )+C$	\dst \int sech\:x \:\dd x= \arctan (\sinh x )+C
$\dst \int csch\:x \:\dd x= arccoth(\cosh x )+C=\ln \tanh (\frac x2 )+C$	\dst \int csch\:x\:\dd x= arccoth(\cosh x )+C= \ln \tanh (\frac x2 )+C
$\dst \int \dfrac 1{x^2+a^2}\:\dd x=\dfrac 1a\arctan \dfrac xa+C=-\dfrac 1a arccot \dfrac xa+C$	\dst \int \dfrac 1{x^2+a^2}\:\dd x= \dfrac 1a\arctan \dfrac xa+C= -\dfrac 1a arccot \dfrac xa+C
$\dst \int \dfrac 1{x^2-a^2}\:\dd x=\dfrac 1{2a}\ln(\dfrac {x-a}{x+a})+C$ $x^2>a^2$	\dst \int \dfrac 1{x^2-a^2}\:\dd x= \dfrac 1{2a}\ln(\dfrac {x-a}{x+a})+C x^2>a^2
$\dst \int \dfrac 1{a^2-x^2}\:\dd x=\dfrac 1{2a}\ln(\dfrac {a+x}{a-x})+C$ $x^2<a^2$	\dst \int \dfrac 1{a^2-x^2}\:\dd x= \dfrac 1{2a}\ln(\dfrac {a+x}{a-x})+C x^2<a^2
$\dst \int \dfrac 1{\sqrt{a^2-x^2}}\:\dd x=\sin \dfrac xa+C=-\cos \dfrac xa+C$	\dst \int \dfrac 1{\sqrt{a^2-x^2}} \:\dd x=\sin \dfrac xa+C= -\cos \dfrac xa+C
$\dst \int \dfrac 1{\sqrt{x^2\pm a^2}}\:\dd x=\ln( x+\sqrt{x^2\pm a^2})+C$	\dst \int \dfrac 1{\sqrt{x^2\pm a^2}} \:\dd x=\ln( x+\sqrt{x^2\pm a^2})+C
$\dst \int \dfrac 1{x\sqrt{a^2 \pm x^2}}\:\dd x=\frac 1a\ln| \dfrac {x}{a+\sqrt{a^2\pm x^2}}|+C$	\dst \int \dfrac 1{x\sqrt{a^2\pm x^2}}\:\dd x=\frac 1a \ln | \dfrac {x}{a+\sqrt{a^2\pm x^2}}|
$\dst \int \dfrac 1{x\sqrt{x^2-a^2}}\:\dd x=\frac 1a\arccos \dfrac {a}{x}=-\frac 1a arcsec \dfrac {x}{a}$	\dst \int \dfrac 1{x\sqrt{x^2-a^2}}\:\dd x= \frac 1a\arccos \dfrac {a}{x}= -\frac 1a arcsec \dfrac {x}{a}+C
$\dst \int \sqrt{a^2-x^2}\:\dd x=\frac x2 \sqrt{a^2\-x^2}+\dfrac {a^2}{2} \arcsin \dfrac {x}{a}+C$	\dst \int \sqrt{a^2-x^2}\:\dd x= \frac x2 \sqrt{a^2-x^2}+\dfrac {a^2} {2} \arcsin \dfrac {x}{a}+C
$\dst \int \sqrt{x^2\pm a^2}\:\dd x=\frac x2 \sqrt{x^2\pm a^2}\pm \dfrac {a^2}{2} \ln (x+\sqrt{x^2\pm a^2})+C$	\dst \int \sqrt{x^2\pm a^2}\:\dd x= \frac x2 \sqrt{x^2\pm a^2}\pm \dfrac {a^2}{2} \ln (x+\sqrt{x^2\pm a^2})+C
$\dst \int e^{ax} \sin bx \:\dd x=\dfrac {e^{ax}(a \sin bx- b\cos bx)}{a^2+b^2}+C$	\dst \int e^{ax} \sin bx \:\dd x= \dfrac {e^{ax}(a \sin bx- b\cos bx)} {a^2+b^2}+C
$\dst \int e^{ax} \cos bx \:\dd x=\dfrac {e^{ax}(a\cos bx+ b \sin bx)}{a^2+b^2}+C$	\dst \int e^{ax} \cos bx \:\dd x= \dfrac {e^{ax}(a\cos bx+ b \sin bx)} {a^2+b^2}+C

Propiedades de Integrales definidas
$\dst \int_a^b (f(x)\pm g(x)) \dd x=\int_a^b f(x) \dd x \pm \int_a^b g(x) \dd x$	\dst \int_a^b (f(x)\pm g(x)) \dd x= \int_a^b f(x)\dd x \pm \int_a^b g(x) \dd x
$\dst \int_a^b C \cdot f(x) \dd x=C \cdot \int_a^b f(x) \dd x$	\dst \int_a^b C \cdot f(x) \dd x= C \cdot \int_a^b f(x) \dd x
$\dst \int_a^b f(x) \dd x=\int_a^c f(x) \dd x + \int_c^b f(x) \dd x$	\dst \int_a^b f(x) \dd x= \int_a^c f(x) \dd x + \int_c^b f(x) \dd x
$\dst \int_a^b f(x) \dd x=-\int_b^a f(x) \dd x$	\dst \int_a^b f(x) \dd x= -\int_b^a f(x) \dd x
$\dst \int_a^a f(x) \dd x=0$	\dst \int_a^a f(x) \dd x=0

Transformadas de Laplace
$F(s)$	F(s)	$f(t)$	f(t)
$\dfrac 1{s}$	\dfrac 1{s}	$1$	1
$\dfrac 1{s^2}$	\dfrac 1{s^2}	$t$	t
$\dfrac 1{s^n}$	\dfrac 1{s^n}	$\dfrac {t^{n-1}}{(n-1)!}$	\dfrac {t^{n-1}}{(n-1)!}
$\dfrac 1{s \pm a}$	\dfrac 1{s \pm a}	$e^{\mp at}$	e^{\mp at}
$\dfrac 1{s(s + a)}$	\dfrac 1{s(s + a)}	$\dfrac 1a(1-e^{-at})$	\dfrac 1a(1-e^{-at})
$\dfrac 1{s^2(s + a)}$	\dfrac 1{s^2(s + a)}	$\dfrac 1{a^2}(e^{-at}+mt-1)$	\dfrac 1{a^2}(e^{-at}+mt-1)
$\dfrac a{s^2 + a^2}$	\dfrac a{s^2 + a^2}	$\sin at$	\sin at
$\dfrac s{s^2 + a^2}$	\dfrac s{s^2 + a^2}	$\cos at$	\cos at
$\dfrac a{s^2 - a^2}$	\dfrac a{s^2 - a^2}	$\sinh at$	\sinh at
$\dfrac s{s^2 - a^2}$	\dfrac s{s^2 - a^2}	$\cosh at$	\cosh at
$\dfrac 1{s(s^2 + a^2)}$	\dfrac 1{s(s^2 + a^2)}	$\dfrac 1{a^2}(1-\cos at)$	\dfrac 1{a^2}(1-\cos at)
$\dfrac 1{s^2(s^2 + a^2)}$	\dfrac 1{s^2(s^2 + a^2)}	$\dfrac 1{a^3}(at-\sin at)$	\dfrac 1{a^3}(at-\sin at)
$\dfrac 1{(s+a) \cdot (s+b)}$	\dfrac 1{(s+a) \cdot (s+b)}	$\dfrac 1{a-b}(e^{-bt}-e^{-at})$	\dfrac 1{a-b}(e^{-bt}-e^{-at})
$\dfrac s{(s+a) \cdot (s+b)}$	\dfrac s{(s+a) \cdot (s+b)}	$\dfrac 1{b-a}(be^{-bt}-ae^{-at})$	\dfrac 1{b-a}(be^{-bt}-ae^{-at})
$\dfrac 1{(s + a)^2}$	\dfrac 1{(s + a)^2}	$te^{- at}$	te^{- at}
$\dfrac 1{(s + a)^n}$	\dfrac 1{(s + a)^n}	$\dfrac {t^{n-1}}{(n-1)!}e^{- at}$	\dfrac {t^{n-1}}{(n-1)!}e^{- at}
$\dfrac s{(s + a)^2}$	\dfrac s{(s + a)^2}	$e^{- at}(1-at)$	e^{- at}(1-at)
$\dfrac 1{(s^2 + a^2)^2}$	\dfrac 1{(s^2 + a^2)^2}	$\dfrac 1{2a^3}(\sin at -at \cos at)$	\dfrac 1{2a^3}(\sin at -at \cos at)
$\dfrac s{(s^2 + a^2)^2}$	\dfrac s{(s^2 + a^2)^2}	$\dfrac t{2a}(\sin at )$	\dfrac t{2a}(\sin at )
$\dfrac {s^2}{(s^2 + a^2)^2}$	\dfrac {s^2}{(s^2 + a^2)^2}	$\dfrac 1{2a}(\sin at +at \cos at)$	\dfrac 1{2a}(\sin at +at \cos at)
$\dfrac {s^2-a^2}{(s^2 + a^2)^2}$	\dfrac {s^2-a^2}{(s^2 + a^2)^2}	$t \cos at$	t \cos at

Propiedades de transformadas de Laplace
Definicion	$\dst F(s)=\mathcal {L} \{ f(t) \}= \int_0^{\infty} e^{-st}f(t) \dd t$	\dst F(s)=\mathcal {L} \{ f(t) \}= \int_0^{\infty} e^{-st}f(t) \dd t
Linealidad	$\mathcal {L} \{ a \cdot f(t) +b \cdot g(t)\}=a \cdot \mathcal {L} \{f(t) \}+b \cdot \mathcal {L} \{ g(t)\}$	\mathcal {L} \{ a \cdot f(t) +b \cdot g(t)\}= a \cdot \mathcal {L} \{f(t) \}+b \cdot \mathcal {L} \{ g(t)\}
Derivación	$\mathcal {L} \{ f'(t) \}=s \mathcal {L} \{ f(t) \}-f(0)$ $\mathcal {L} \{ f''(t) \}=s^2 \mathcal {L} \{ f(t) \}-s f(0)-f'(0)$ $\mathcal {L} \{ f^{(n)}(t) \}=s^n \mathcal {L} \{ f(t) \}-s^{n-1} f(0)-\ldots-f^{(n-1)}(0)$ $\mathcal {L} \{ f^{(n)}(t) \}=s^n \mathcal {L} \{ f(t) \}-\dst \sum_{i=1}^n s^{n-i}f^{(i-1)}(0)$	\mathcal {L} \{ f'(t) \}=s \mathcal {L} \{ f(t) \}-f(0) \mathcal {L} \{ f''(t) \}=s^2 \mathcal {L} \{ f(t) \}-s f(0)-f'(0) \mathcal {L} \{ f^{(n)}(t) \}= s^n \mathcal {L} \{ f(t) \}-s^{n-1} f(0)-\ldots-f^{(n-1)}(0) \mathcal {L} \{ f^{(n)}(t) \}= s^n \mathcal {L} \{ f(t) \}-\dst \sum_{i=1}^n s^{n-i}f^{(i-1)}(0)
Integración	$\mathcal {L} \{ \int_{0-}^t f(\tau) \dd \tau \}=\dfrac 1s \mathcal {L} \{ f(t) \}$	\mathcal {L} \{ \int_{0-}^t f(\tau) \dd \tau \}=\dfrac 1s \mathcal {L} \{ f(t) \}
Dualidad	$\mathcal {L} \{ t \cdot f(t) \}=-F'(s)$	\mathcal {L} \{ t \cdot f(t) \}=-F'(s)
Desplazamiento de frecuencia	$\mathcal {L} \{ e^{at}f(t) \}=F(s-a)$	\mathcal {L} \{ e^{at}f(t) \}=F(s-a)
Desplazamiento temporal	$\mathcal {L} \{ f(t-a) \cdot u(t-a) \}=e^{-as} F(s)$ $\mathcal {L}^{-1} \{ e^{-as} F(s) \}=f(t-a) \cdot u(t-a)$ $u(t-a)=\left\{ \begin{array}{cl}0 & \text{si}\ t < a \\<br /> 1 & \text{si}\ t \ge a<br /> \end{array}\right .$	\mathcal {L} \{ f(t-a) \cdot u(t-a) \}=e^{-as} F(s) \mathcal {L}^{-1} \{ e^{-as} F(s) \}=f(t-a) \cdot u(t-a) u(t-a)=\left\{ \begin{array}{cl}0 & \text{si}\ t < a \\ 1 & \text{si}\ t \ge a \end{array}\right.
Convolucion	$\mathcal {L} \{ f(t) \cdot g(t) \}=F(s) \cdot G(s)$	\mathcal {L} \{ f(t) \cdot g(t) \}=F(s) \cdot G(s)

Si halla alguna errata o desea colaborar indicando alguna omisión les agradeceré que las comenten y las corregiré a la brevedad.