Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
exp(x)/(E^x-E^(-x))+(E^x+1)*(-E^x-exp(-x))/(E^x-E^(-x))^2
¿Cómo vas a descomponer esta exp(x)/(E^x-E^(-x))+(E^x+1)*(-E^x-exp(-x))/(E^x-E^(-x))^2 expresión en fracciones?
Solución
Ha introducido
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x / x \ / x -x\
e \E + 1/*\- E - e /
-------- + ---------------------
x -x 2
E - E / x -x\
\E - E /
$$\frac{\left(- e^{x} - e^{- x}\right) \left(e^{x} + 1\right)}{\left(e^{x} - e^{- x}\right)^{2}} + \frac{e^{x}}{e^{x} - e^{- x}}$$
exp(x)/(E^x - E^(-x)) + ((E^x + 1)*(-E^x - exp(-x)))/(E^x - E^(-x))^2
Descomposición de una fracción
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-1/(-1 + exp(x)) - 1/(-1 + exp(x))^2
$$- \frac{1}{e^{x} - 1} - \frac{1}{\left(e^{x} - 1\right)^{2}}$$
1 1
- ------- - ----------
x 2
-1 + e / x\
\-1 + e /
Simplificación general
[src]
x
-e
---------------
x 2*x
1 - 2*e + e
$$- \frac{e^{x}}{e^{2 x} - 2 e^{x} + 1}$$
-exp(x)/(1 - 2*exp(x) + exp(2*x))
Combinatoria
[src]
x
-e
----------
2
/ x\
\-1 + e /
$$- \frac{e^{x}}{\left(e^{x} - 1\right)^{2}}$$
-exp(x)/(-1 + exp(x))^2
Denominador común
[src]
x
-e
---------------
x 2*x
1 - 2*e + e
$$- \frac{e^{x}}{e^{2 x} - 2 e^{x} + 1}$$
-exp(x)/(1 - 2*exp(x) + exp(2*x))
Respuesta numérica
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exp(x)/(2.71828182845905^x - 2.71828182845905^(-x)) + (1.0 + 2.71828182845905^x)*(-2.71828182845905^x - exp(-x))/(2.71828182845905^x - 2.71828182845905^(-x))^2
exp(x)/(2.71828182845905^x - 2.71828182845905^(-x)) + (1.0 + 2.71828182845905^x)*(-2.71828182845905^x - exp(-x))/(2.71828182845905^x - 2.71828182845905^(-x))^2
Unión de expresiones racionales
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// x\ / 2*x\ / 2*x\ x\ x
\\1 + e /*\-1 - e / + \-1 + e /*e /*e
------------------------------------------
2
/ 2*x\
\-1 + e /
$$\frac{\left(\left(e^{x} + 1\right) \left(- e^{2 x} - 1\right) + \left(e^{2 x} - 1\right) e^{x}\right) e^{x}}{\left(e^{2 x} - 1\right)^{2}}$$
((1 + exp(x))*(-1 - exp(2*x)) + (-1 + exp(2*x))*exp(x))*exp(x)/(-1 + exp(2*x))^2
Denominador racional
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/ 2 \
|/ 2*x\ 3*x / x\ / 2*x\ / 2*x\ 2*x| -x
\\-1 + e / *e + \1 + e /*\-1 - e /*\-1 + e /*e /*e
---------------------------------------------------------------
3
/ 2*x\
\-1 + e /
$$\frac{\left(\left(e^{x} + 1\right) \left(- e^{2 x} - 1\right) \left(e^{2 x} - 1\right) e^{2 x} + \left(e^{2 x} - 1\right)^{2} e^{3 x}\right) e^{- x}}{\left(e^{2 x} - 1\right)^{3}}$$
((-1 + exp(2*x))^2*exp(3*x) + (1 + exp(x))*(-1 - exp(2*x))*(-1 + exp(2*x))*exp(2*x))*exp(-x)/(-1 + exp(2*x))^3
Parte trigonométrica
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cosh(x) + sinh(x) (1 + cosh(x) + sinh(x))*cosh(x)
----------------- - -------------------------------
2*sinh(x) 2
2*sinh (x)
$$\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{2 \sinh{\left(x \right)}} - \frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right) \cosh{\left(x \right)}}{2 \sinh^{2}{\left(x \right)}}$$
x / 2*x x \
-(cosh(1) + sinh(1)) *\(cosh(1) + sinh(1)) - 2*(cosh(1) + sinh(1)) *sinh(x) + cosh(x) + sinh(x)/
----------------------------------------------------------------------------------------------------
2
/ x\ / x\
\1 + (cosh(1) + sinh(1)) /*\-1 + (cosh(1) + sinh(1)) /
$$- \frac{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} \left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{2 x} - 2 \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} \sinh{\left(x \right)} + \sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - 1\right)^{2} \left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 1\right)}$$
-(1 + cosh(x))
---------------
2
2*sinh (x)
$$- \frac{\cosh{\left(x \right)} + 1}{2 \sinh^{2}{\left(x \right)}}$$
/ x\ / x \
cosh(x) + sinh(x) \1 + (cosh(1) + sinh(1)) /*\- (cosh(1) + sinh(1)) - cosh(x) + sinh(x)/
-------------------------------------------- + -----------------------------------------------------------------------
x -x 2
(cosh(1) + sinh(1)) - (cosh(1) + sinh(1)) / x -x\
\(cosh(1) + sinh(1)) - (cosh(1) + sinh(1)) /
$$\frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 1\right) \left(- \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + \sinh{\left(x \right)} - \cosh{\left(x \right)}\right)}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- x}\right)^{2}} + \frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- x}}$$
(cosh(x) + sinh(x))/((cosh(1) + sinh(1))^x - (cosh(1) + sinh(1))^(-x)) + (1 + (cosh(1) + sinh(1))^x)*(-(cosh(1) + sinh(1))^x - cosh(x) + sinh(x))/((cosh(1) + sinh(1))^x - (cosh(1) + sinh(1))^(-x))^2