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