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Simplificación de expresiones/ Descomposición en fracciones simples/ sqrt(x/(x-1))*(x-1)*(1/(2*(x-1))-x/(2*(x-1)^2))/(x*sqrt(1-x/(x-1)))

¿Cómo vas a descomponer esta sqrt(x/(x-1))*(x-1)*(1/(2*(x-1))-x/(2*(x-1)^2))/(x*sqrt(1-x/(x-1))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    _______                                 
   /   x            /    1           x     \
  /  ----- *(x - 1)*|--------- - ----------|
\/   x - 1          |2*(x - 1)            2|
                    \            2*(x - 1) /
--------------------------------------------
                   ___________              
                  /       x                 
             x*  /  1 - -----               
               \/       x - 1
xx−1(x−1)(−x2(x−1)2+12(x−1))x−xx−1+1\frac{\sqrt{\frac{x}{x - 1}} \left(x - 1\right) \left(- \frac{x}{2 \left(x - 1\right)^{2}} + \frac{1}{2 \left(x - 1\right)}\right)}{x \sqrt{- \frac{x}{x - 1} + 1}} 
((sqrt(x/(x - 1))*(x - 1))*(1/(2*(x - 1)) - x/(2*(x - 1)^2)))/((x*sqrt(1 - x/(x - 1))))
Simplificación general [src] 
           ________      
          /   x          
      -  /  ------       
       \/   -1 + x       
-------------------------
        ________         
       /  -1             
2*x*  /  ------ *(-1 + x)
    \/   -1 + x
−xx−12x−1x−1(x−1)- \frac{\sqrt{\frac{x}{x - 1}}}{2 x \sqrt{- \frac{1}{x - 1}} \left(x - 1\right)} 
-sqrt(x/(-1 + x))/(2*x*sqrt(-1/(-1 + x))*(-1 + x))
Combinatoria [src] 
           ________      
          /   x          
      -  /  ------       
       \/   -1 + x       
-------------------------
        ________         
       /  -1             
2*x*  /  ------ *(-1 + x)
    \/   -1 + x
−xx−12x−1x−1(x−1)- \frac{\sqrt{\frac{x}{x - 1}}}{2 x \sqrt{- \frac{1}{x - 1}} \left(x - 1\right)} 
-sqrt(x/(-1 + x))/(2*x*sqrt(-1/(-1 + x))*(-1 + x))
Potencias [src] 
    ________                                  
   /   x              /   1            x     \
  /  ------ *(-1 + x)*|-------- - -----------|
\/   -1 + x           |-2 + 2*x             2|
                      \           2*(-1 + x) /
----------------------------------------------
                    ____________              
                   /       x                  
              x*  /  1 - ------               
                \/       -1 + x
xx−1(x−1)(−x2(x−1)2+12x−2)x−xx−1+1\frac{\sqrt{\frac{x}{x - 1}} \left(x - 1\right) \left(- \frac{x}{2 \left(x - 1\right)^{2}} + \frac{1}{2 x - 2}\right)}{x \sqrt{- \frac{x}{x - 1} + 1}} 
sqrt(x/(-1 + x))*(-1 + x)*(1/(-2 + 2*x) - x/(2*(-1 + x)^2))/(x*sqrt(1 - x/(-1 + x)))
Respuesta numérica [src] 
(x/(-1.0 + x))^0.5*(1.0 - x/(-1.0 + x))^(-0.5)*(-1.0 + x)*(1/(-2.0 + 2.0*x) - 0.5*x/(-1.0 + x)^2)/x
(x/(-1.0 + x))^0.5*(1.0 - x/(-1.0 + x))^(-0.5)*(-1.0 + x)*(1/(-2.0 + 2.0*x) - 0.5*x/(-1.0 + x)^2)/x
Unión de expresiones racionales [src] 
           ________      
          /   x          
      -  /  ------       
       \/   -1 + x       
-------------------------
        ________         
       /  -1             
2*x*  /  ------ *(-1 + x)
    \/   -1 + x
−xx−12x−1x−1(x−1)- \frac{\sqrt{\frac{x}{x - 1}}}{2 x \sqrt{- \frac{1}{x - 1}} \left(x - 1\right)} 
-sqrt(x/(-1 + x))/(2*x*sqrt(-1/(-1 + x))*(-1 + x))
Denominador común [src] 
    ________     ____________
   /   x        /       x    
  /  ------ *  /  1 - ------ 
\/   -1 + x  \/       -1 + x 
-----------------------------
             2*x
xx−1−xx−1+12x\frac{\sqrt{\frac{x}{x - 1}} \sqrt{- \frac{x}{x - 1} + 1}}{2 x} 
sqrt(x/(-1 + x))*sqrt(1 - x/(-1 + x))/(2*x)
Abrimos la expresión [src] 
    _______                                 
   /   1            /    1           x     \
  /  ----- *(x - 1)*|--------- - ----------|
\/   x - 1          |2*(x - 1)            2|
                    \            2*(x - 1) /
--------------------------------------------
                     ___________            
             ___    /       x               
           \/ x *  /  1 - -----             
                 \/       x - 1
(x−1)(−x2(x−1)2+12(x−1))1x−1x−xx−1+1\frac{\left(x - 1\right) \left(- \frac{x}{2 \left(x - 1\right)^{2}} + \frac{1}{2 \left(x - 1\right)}\right) \sqrt{\frac{1}{x - 1}}}{\sqrt{x} \sqrt{- \frac{x}{x - 1} + 1}} 
sqrt(1/(x - 1))*(x - 1)*(1/(2*(x - 1)) - x/(2*(x - 1)^2))/(sqrt(x)*sqrt(1 - x/(x - 1)))
Compilar la expresión [src] 
    ________                                  
   /   x              /   1            x     \
  /  ------ *(-1 + x)*|-------- - -----------|
\/   -1 + x           |-2 + 2*x             2|
                      \           2*(-1 + x) /
----------------------------------------------
                    ____________              
                   /       x                  
              x*  /  1 - ------               
                \/       -1 + x
xx−1(x−1)(−x2(x−1)2+12x−2)x−xx−1+1\frac{\sqrt{\frac{x}{x - 1}} \left(x - 1\right) \left(- \frac{x}{2 \left(x - 1\right)^{2}} + \frac{1}{2 x - 2}\right)}{x \sqrt{- \frac{x}{x - 1} + 1}} 
sqrt(x/(-1 + x))*(-1 + x)*(1/(-2 + 2*x) - x/(2*(-1 + x)^2))/(x*sqrt(1 - x/(-1 + x)))
Parte trigonométrica [src] 
    ________                                  
   /   x              /   1            x     \
  /  ------ *(-1 + x)*|-------- - -----------|
\/   -1 + x           |-2 + 2*x             2|
                      \           2*(-1 + x) /
----------------------------------------------
                    ____________              
                   /       x                  
              x*  /  1 - ------               
                \/       -1 + x
xx−1(x−1)(−x2(x−1)2+12x−2)x−xx−1+1\frac{\sqrt{\frac{x}{x - 1}} \left(x - 1\right) \left(- \frac{x}{2 \left(x - 1\right)^{2}} + \frac{1}{2 x - 2}\right)}{x \sqrt{- \frac{x}{x - 1} + 1}} 
sqrt(x/(-1 + x))*(-1 + x)*(1/(-2 + 2*x) - x/(2*(-1 + x)^2))/(x*sqrt(1 - x/(-1 + x)))
Denominador racional [src] 
          _______     ____________         _______     ____________                      _______     ____________
         /   x       /       x            /   x       /       x             2      2    /   x       /       x    
- 2*x*  /  ----- *  /  1 - ------  - 2*  /  ----- *  /  1 - ------ *(-1 + x)  + 2*x *  /  ----- *  /  1 - ------ 
      \/   x - 1  \/       -1 + x      \/   x - 1  \/       -1 + x                   \/   x - 1  \/       -1 + x 
-----------------------------------------------------------------------------------------------------------------
                                                   /       x   \                                                 
                                      2*x*(-1 + x)*|-1 + ------|*(-2 + 2*x)                                      
                                                   \     -1 + x/
2x2xx−1−xx−1+1−2xxx−1−xx−1+1−2xx−1(x−1)2−xx−1+12x(x−1)(2x−2)(xx−1−1)\frac{2 x^{2} \sqrt{\frac{x}{x - 1}} \sqrt{- \frac{x}{x - 1} + 1} - 2 x \sqrt{\frac{x}{x - 1}} \sqrt{- \frac{x}{x - 1} + 1} - 2 \sqrt{\frac{x}{x - 1}} \left(x - 1\right)^{2} \sqrt{- \frac{x}{x - 1} + 1}}{2 x \left(x - 1\right) \left(2 x - 2\right) \left(\frac{x}{x - 1} - 1\right)} 
(-2*x*sqrt(x/(x - 1))*sqrt(1 - x/(-1 + x)) - 2*sqrt(x/(x - 1))*sqrt(1 - x/(-1 + x))*(-1 + x)^2 + 2*x^2*sqrt(x/(x - 1))*sqrt(1 - x/(-1 + x)))/(2*x*(-1 + x)*(-1 + x/(-1 + x))*(-2 + 2*x))
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