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Simplificación de expresiones/ Descomposición en fracciones simples/ exp(x)/(e^x-e^3)-exp(2*x)/(e^x-e^3)^2

¿Cómo vas a descomponer esta exp(x)/(e^x-e^3)-exp(2*x)/(e^x-e^3)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    x         2*x   
   e         e      
------- - ----------
 x    3            2
E  - E    / x    3\ 
          \E  - E /
−e2x(ex−e3)2+exex−e3- \frac{e^{2 x}}{\left(e^{x} - e^{3}\right)^{2}} + \frac{e^{x}}{e^{x} - e^{3}} 
exp(x)/(E^x - E^3) - exp(2*x)/(E^x - E^3)^2
Simplificación general [src] 
         3 + x        
       -e             
----------------------
     3 + x    6    2*x
- 2*e      + e  + e
−ex+3e2x−2ex+3+e6- \frac{e^{x + 3}}{e^{2 x} - 2 e^{x + 3} + e^{6}} 
-exp(3 + x)/(-2*exp(3 + x) + exp(6) + exp(2*x))
Respuesta numérica [src] 
exp(x)/(-20.0855369231877 + 2.71828182845905^x) - 0.00247875217666636*exp(2*x)/(-1 + 0.049787068367864*2.71828182845905^x)^2
exp(x)/(-20.0855369231877 + 2.71828182845905^x) - 0.00247875217666636*exp(2*x)/(-1 + 0.049787068367864*2.71828182845905^x)^2
Denominador racional [src] 
           2                      
/   3    x\   x   /   3    x\  2*x
\- e  + e / *e  - \- e  + e /*e   
----------------------------------
                      3           
           /   3    x\            
           \- e  + e /
(ex−e3)2ex−(ex−e3)e2x(ex−e3)3\frac{\left(e^{x} - e^{3}\right)^{2} e^{x} - \left(e^{x} - e^{3}\right) e^{2 x}}{\left(e^{x} - e^{3}\right)^{3}} 
((-exp(3) + exp(x))^2*exp(x) - (-exp(3) + exp(x))*exp(2*x))/(-exp(3) + exp(x))^3
Compilar la expresión [src] 
     x           2*x    
    e           e       
--------- - ------------
   3    x              2
- e  + e    /   3    x\ 
            \- e  + e /
exex−e3−e2x(ex−e3)2\frac{e^{x}}{e^{x} - e^{3}} - \frac{e^{2 x}}{\left(e^{x} - e^{3}\right)^{2}} 
exp(x)/(-exp(3) + exp(x)) - exp(2*x)/(-exp(3) + exp(x))^2
Unión de expresiones racionales [src] 
    3  x    
  -e *e     
------------
           2
/   3    x\ 
\- e  + e /
−e3ex(ex−e3)2- \frac{e^{3} e^{x}}{\left(e^{x} - e^{3}\right)^{2}} 
-exp(3)*exp(x)/(-exp(3) + exp(x))^2
Combinatoria [src] 
    3  x    
  -e *e     
------------
           2
/   x    3\ 
\- e  + e /
−e3ex(−ex+e3)2- \frac{e^{3} e^{x}}{\left(- e^{x} + e^{3}\right)^{2}} 
-exp(3)*exp(x)/(-exp(x) + exp(3))^2
Potencias [src] 
     x           2*x    
    e           e       
--------- - ------------
   3    x              2
- e  + e    /   3    x\ 
            \- e  + e /
exex−e3−e2x(ex−e3)2\frac{e^{x}}{e^{x} - e^{3}} - \frac{e^{2 x}}{\left(e^{x} - e^{3}\right)^{2}} 
exp(x)/(-exp(3) + exp(x)) - exp(2*x)/(-exp(3) + exp(x))^2
Denominador común [src] 
         3  x        
       -e *e         
---------------------
     3  x    6    2*x
- 2*e *e  + e  + e
−e3exe2x−2e3ex+e6- \frac{e^{3} e^{x}}{e^{2 x} - 2 e^{3} e^{x} + e^{6}} 
-exp(3)*exp(x)/(-2*exp(3)*exp(x) + exp(6) + exp(2*x))
Parte trigonométrica [src] 
      -(cosh(3 + x) + sinh(3 + x))       
-----------------------------------------
                                        2
(-cosh(3) - sinh(3) + cosh(x) + sinh(x))
−sinh⁡(x+3)+cosh⁡(x+3)(sinh⁡(x)+cosh⁡(x)−cosh⁡(3)−sinh⁡(3))2- \frac{\sinh{\left(x + 3 \right)} + \cosh{\left(x + 3 \right)}}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} - \cosh{\left(3 \right)} - \sinh{\left(3 \right)}\right)^{2}} 
             cosh(x) + sinh(x)                            cosh(2*x) + sinh(2*x)             
------------------------------------------- - ----------------------------------------------
                   x                      3                                                2
(cosh(1) + sinh(1))  - (cosh(1) + sinh(1))    /                   x                      3\ 
                                              \(cosh(1) + sinh(1))  - (cosh(1) + sinh(1)) /
sinh⁡(x)+cosh⁡(x)(sinh⁡(1)+cosh⁡(1))x−(sinh⁡(1)+cosh⁡(1))3−sinh⁡(2x)+cosh⁡(2x)((sinh⁡(1)+cosh⁡(1))x−(sinh⁡(1)+cosh⁡(1))3)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)^{3}} - \frac{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)}}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{3}\right)^{2}} 
          cosh(x) + sinh(x)                        cosh(2*x) + sinh(2*x)          
-------------------------------------- - -----------------------------------------
-cosh(3) - sinh(3) + cosh(x) + sinh(x)                                           2
                                         (-cosh(3) - sinh(3) + cosh(x) + sinh(x))
sinh⁡(x)+cosh⁡(x)sinh⁡(x)+cosh⁡(x)−cosh⁡(3)−sinh⁡(3)−sinh⁡(2x)+cosh⁡(2x)(sinh⁡(x)+cosh⁡(x)−cosh⁡(3)−sinh⁡(3))2\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\sinh{\left(x \right)} + \cosh{\left(x \right)} - \cosh{\left(3 \right)} - \sinh{\left(3 \right)}} - \frac{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)}}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} - \cosh{\left(3 \right)} - \sinh{\left(3 \right)}\right)^{2}} 
     x           2*x    
    e           e       
--------- - ------------
   3    x              2
- e  + e    /   3    x\ 
            \- e  + e /
exex−e3−e2x(ex−e3)2\frac{e^{x}}{e^{x} - e^{3}} - \frac{e^{2 x}}{\left(e^{x} - e^{3}\right)^{2}} 
exp(x)/(-exp(3) + exp(x)) - exp(2*x)/(-exp(3) + exp(x))^2
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