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Simplificación de expresiones/ Descomposición en fracciones simples/ exp(x)/(E^x-2)-exp(2*x)/(E^x-2)^2

¿Cómo vas a descomponer esta exp(x)/(E^x-2)-exp(2*x)/(E^x-2)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   x         2*x  
  e         e     
------ - ---------
 x               2
E  - 2   / x    \ 
         \E  - 2/
−e2x(ex−2)2+exex−2- \frac{e^{2 x}}{\left(e^{x} - 2\right)^{2}} + \frac{e^{x}}{e^{x} - 2} 
exp(x)/(E^x - 2) - exp(2*x)/(E^x - 2)^2
Descomposición de una fracción [src] 
-4/(-2 + exp(x))^2 - 2/(-2 + exp(x))
−2ex−2−4(ex−2)2- \frac{2}{e^{x} - 2} - \frac{4}{\left(e^{x} - 2\right)^{2}} 
      4           2   
- ---------- - -------
           2         x
  /      x\    -2 + e 
  \-2 + e /
Simplificación general [src] 
         x     
     -2*e      
---------------
       x    2*x
4 - 4*e  + e
−2exe2x−4ex+4- \frac{2 e^{x}}{e^{2 x} - 4 e^{x} + 4} 
-2*exp(x)/(4 - 4*exp(x) + exp(2*x))
Respuesta numérica [src] 
exp(x)/(-2.0 + 2.71828182845905^x) - 0.25*exp(2*x)/(-1 + 0.5*2.71828182845905^x)^2
exp(x)/(-2.0 + 2.71828182845905^x) - 0.25*exp(2*x)/(-1 + 0.5*2.71828182845905^x)^2
Denominador común [src] 
         x     
     -2*e      
---------------
       x    2*x
4 - 4*e  + e
−2exe2x−4ex+4- \frac{2 e^{x}}{e^{2 x} - 4 e^{x} + 4} 
-2*exp(x)/(4 - 4*exp(x) + exp(2*x))
Compilar la expresión [src] 
    x         2*x   
   e         e      
------- - ----------
      x            2
-2 + e    /      x\ 
          \-2 + e /
exex−2−e2x(ex−2)2\frac{e^{x}}{e^{x} - 2} - \frac{e^{2 x}}{\left(e^{x} - 2\right)^{2}} 
exp(x)/(-2 + exp(x)) - exp(2*x)/(-2 + exp(x))^2
Potencias [src] 
    x         2*x   
   e         e      
------- - ----------
      x            2
-2 + e    /      x\ 
          \-2 + e /
exex−2−e2x(ex−2)2\frac{e^{x}}{e^{x} - 2} - \frac{e^{2 x}}{\left(e^{x} - 2\right)^{2}} 
exp(x)/(-2 + exp(x)) - exp(2*x)/(-2 + exp(x))^2
Denominador racional [src] 
         2                    
/      x\   x   /      x\  2*x
\-2 + e / *e  - \-2 + e /*e   
------------------------------
                   3          
          /      x\           
          \-2 + e /
(ex−2)2ex−(ex−2)e2x(ex−2)3\frac{\left(e^{x} - 2\right)^{2} e^{x} - \left(e^{x} - 2\right) e^{2 x}}{\left(e^{x} - 2\right)^{3}} 
((-2 + exp(x))^2*exp(x) - (-2 + exp(x))*exp(2*x))/(-2 + exp(x))^3
Combinatoria [src] 
      x   
  -2*e    
----------
         2
/      x\ 
\-2 + e /
−2ex(ex−2)2- \frac{2 e^{x}}{\left(e^{x} - 2\right)^{2}} 
-2*exp(x)/(-2 + exp(x))^2
Parte trigonométrica [src] 
  cosh(x) + sinh(x)        cosh(2*x) + sinh(2*x)  
---------------------- - -------------------------
-2 + cosh(x) + sinh(x)                           2
                         (-2 + cosh(x) + sinh(x))
sinh⁡(x)+cosh⁡(x)sinh⁡(x)+cosh⁡(x)−2−sinh⁡(2x)+cosh⁡(2x)(sinh⁡(x)+cosh⁡(x)−2)2\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\sinh{\left(x \right)} + \cosh{\left(x \right)} - 2} - \frac{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)}}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} - 2\right)^{2}} 
    x         2*x   
   e         e      
------- - ----------
      x            2
-2 + e    /      x\ 
          \-2 + e /
exex−2−e2x(ex−2)2\frac{e^{x}}{e^{x} - 2} - \frac{e^{2 x}}{\left(e^{x} - 2\right)^{2}} 
    cosh(x) + sinh(x)          cosh(2*x) + sinh(2*x)    
------------------------- - ----------------------------
                        x                              2
-2 + (cosh(1) + sinh(1))    /                        x\ 
                            \-2 + (cosh(1) + sinh(1)) /
sinh⁡(x)+cosh⁡(x)(sinh⁡(1)+cosh⁡(1))x−2−sinh⁡(2x)+cosh⁡(2x)((sinh⁡(1)+cosh⁡(1))x−2)2\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - 2} - \frac{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)}}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - 2\right)^{2}} 
  -2*(cosh(x) + sinh(x)) 
-------------------------
                        2
(-2 + cosh(x) + sinh(x))
−2(sinh⁡(x)+cosh⁡(x))(sinh⁡(x)+cosh⁡(x)−2)2- \frac{2 \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} - 2\right)^{2}} 
-2*(cosh(x) + sinh(x))/(-2 + cosh(x) + sinh(x))^2
Unión de expresiones racionales [src] 
      x   
  -2*e    
----------
         2
/      x\ 
\-2 + e /
−2ex(ex−2)2- \frac{2 e^{x}}{\left(e^{x} - 2\right)^{2}} 
-2*exp(x)/(-2 + exp(x))^2
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