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Simplificación de expresiones/ Descomposición en fracciones simples/ sqrt(x-x^2)/2+1/(4*sqrt(1-(2*x-1)^2))+(1/2-x)*(2*x-1)/(4*sqrt(x-x^2))

¿Cómo vas a descomponer esta sqrt(x-x^2)/2+1/(4*sqrt(1-(2*x-1)^2))+(1/2-x)*(2*x-1)/(4*sqrt(x-x^2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   ________                                              
  /      2                                               
\/  x - x               1             (1/2 - x)*(2*x - 1)
----------- + --------------------- + -------------------
     2             ________________           ________   
                  /              2           /      2    
              4*\/  1 - (2*x - 1)        4*\/  x - x
(12−x)(2x−1)4−x2+x+(−x2+x2+141−(2x−1)2)\frac{\left(\frac{1}{2} - x\right) \left(2 x - 1\right)}{4 \sqrt{- x^{2} + x}} + \left(\frac{\sqrt{- x^{2} + x}}{2} + \frac{1}{4 \sqrt{1 - \left(2 x - 1\right)^{2}}}\right) 
sqrt(x - x^2)/2 + 1/(4*sqrt(1 - (2*x - 1)^2)) + ((1/2 - x)*(2*x - 1))/((4*sqrt(x - x^2)))
Descomposición de una fracción [src] 
sqrt(x - x^2)
−x2+x\sqrt{- x^{2} + x} 
   ________
  /      2 
\/  x - x
Simplificación general [src] 
  -x*(-1 + x)  
---------------
  _____________
\/ -x*(-1 + x)
−x(x−1)−x(x−1)- \frac{x \left(x - 1\right)}{\sqrt{- x \left(x - 1\right)}} 
-x*(-1 + x)/sqrt(-x*(-1 + x))
Potencias [src] 
   ________                                                
  /      2                                                 
\/  x - x               1              (1/2 - x)*(-1 + 2*x)
----------- + ---------------------- + --------------------
     2             _________________           ________    
                  /               2           /      2     
              4*\/  1 - (-1 + 2*x)        4*\/  x - x
(12−x)(2x−1)4−x2+x+−x2+x2+141−(2x−1)2\frac{\left(\frac{1}{2} - x\right) \left(2 x - 1\right)}{4 \sqrt{- x^{2} + x}} + \frac{\sqrt{- x^{2} + x}}{2} + \frac{1}{4 \sqrt{1 - \left(2 x - 1\right)^{2}}} 
sqrt(x - x^2)/2 + 1/(4*sqrt(1 - (-1 + 2*x)^2)) + (1/2 - x)*(-1 + 2*x)/(4*sqrt(x - x^2))
Respuesta numérica [src] 
0.125*(0.25 - (-0.5 + x)^2)^(-0.5) + 0.5*(x - x^2)^0.5 + 0.25*(x - x^2)^(-0.5)*(0.5 - x)*(-1.0 + 2.0*x)
0.125*(0.25 - (-0.5 + x)^2)^(-0.5) + 0.5*(x - x^2)^0.5 + 0.25*(x - x^2)^(-0.5)*(0.5 - x)*(-1.0 + 2.0*x)
Parte trigonométrica [src] 
   ________                                                
  /      2                                                 
\/  x - x               1              (1/2 - x)*(-1 + 2*x)
----------- + ---------------------- + --------------------
     2             _________________           ________    
                  /               2           /      2     
              4*\/  1 - (-1 + 2*x)        4*\/  x - x
(12−x)(2x−1)4−x2+x+−x2+x2+141−(2x−1)2\frac{\left(\frac{1}{2} - x\right) \left(2 x - 1\right)}{4 \sqrt{- x^{2} + x}} + \frac{\sqrt{- x^{2} + x}}{2} + \frac{1}{4 \sqrt{1 - \left(2 x - 1\right)^{2}}} 
sqrt(x - x^2)/2 + 1/(4*sqrt(1 - (-1 + 2*x)^2)) + (1/2 - x)*(-1 + 2*x)/(4*sqrt(x - x^2))
Abrimos la expresión [src] 
                           ________                      
                          /      2                       
          1             \/  x - x     (1/2 - x)*(2*x - 1)
--------------------- + ----------- + -------------------
     ________________        2                ________   
    /              2                         /      2    
4*\/  1 - (2*x - 1)                      4*\/  x - x
(12−x)(2x−1)4−x2+x+−x2+x2+141−(2x−1)2\frac{\left(\frac{1}{2} - x\right) \left(2 x - 1\right)}{4 \sqrt{- x^{2} + x}} + \frac{\sqrt{- x^{2} + x}}{2} + \frac{1}{4 \sqrt{1 - \left(2 x - 1\right)^{2}}} 
1/(4*sqrt(1 - (2*x - 1)^2)) + sqrt(x - x^2)/2 + (1/2 - x)*(2*x - 1)/(4*sqrt(x - x^2))
Denominador racional [src] 
           ______________           ______________         _________________         3/2    ______________        _________________                ________    ______________
      2   /      2                 /      2               /               2  /     2\      /      2              /               2            2   /      2    /      2       
- 16*x *\/  - 4*x  + 4*x  + 16*x*\/  - 4*x  + 4*x  + 32*\/  1 - (-1 + 2*x)  *\x - x /   *\/  - 4*x  + 4*x  - 8*\/  1 - (-1 + 2*x)  *(-1 + 2*x) *\/  x - x  *\/  - 4*x  + 4*x 
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                                                                             / 2    \ /          2\                                                                          
                                                                          64*\x  - x/*\-4*x + 4*x /
−16x2−4x2+4x+16x−4x2+4x−81−(2x−1)2(2x−1)2−4x2+4x−x2+x+321−(2x−1)2−4x2+4x(−x2+x)3264(x2−x)(4x2−4x)\frac{- 16 x^{2} \sqrt{- 4 x^{2} + 4 x} + 16 x \sqrt{- 4 x^{2} + 4 x} - 8 \sqrt{1 - \left(2 x - 1\right)^{2}} \left(2 x - 1\right)^{2} \sqrt{- 4 x^{2} + 4 x} \sqrt{- x^{2} + x} + 32 \sqrt{1 - \left(2 x - 1\right)^{2}} \sqrt{- 4 x^{2} + 4 x} \left(- x^{2} + x\right)^{\frac{3}{2}}}{64 \left(x^{2} - x\right) \left(4 x^{2} - 4 x\right)} 
(-16*x^2*sqrt(-4*x^2 + 4*x) + 16*x*sqrt(-4*x^2 + 4*x) + 32*sqrt(1 - (-1 + 2*x)^2)*(x - x^2)^(3/2)*sqrt(-4*x^2 + 4*x) - 8*sqrt(1 - (-1 + 2*x)^2)*(-1 + 2*x)^2*sqrt(x - x^2)*sqrt(-4*x^2 + 4*x))/(64*(x^2 - x)*(-4*x + 4*x^2))
Compilar la expresión [src] 
   ________                                                
  /      2                                                 
\/  x - x               1              (1/2 - x)*(-1 + 2*x)
----------- + ---------------------- + --------------------
     2             _________________           ________    
                  /               2           /      2     
              4*\/  1 - (-1 + 2*x)        4*\/  x - x
(12−x)(2x−1)4−x2+x+−x2+x2+141−(2x−1)2\frac{\left(\frac{1}{2} - x\right) \left(2 x - 1\right)}{4 \sqrt{- x^{2} + x}} + \frac{\sqrt{- x^{2} + x}}{2} + \frac{1}{4 \sqrt{1 - \left(2 x - 1\right)^{2}}} 
sqrt(x - x^2)/2 + 1/(4*sqrt(1 - (-1 + 2*x)^2)) + (1/2 - x)*(-1 + 2*x)/(4*sqrt(x - x^2))
Denominador común [src] 
   ________
  /      2 
\/  x - x
−x2+x\sqrt{- x^{2} + x} 
sqrt(x - x^2)
Combinatoria [src] 
  _____________
\/ -x*(-1 + x)
−x(x−1)\sqrt{- x \left(x - 1\right)} 
sqrt(-x*(-1 + x))
Unión de expresiones racionales [src] 
                /                       _________________\      _________________                     
    ___________ |        ___________   /               2 |     /               2                      
2*\/ x*(1 - x) *\1 + 2*\/ x*(1 - x) *\/  1 - (-1 + 2*x)  / + \/  1 - (-1 + 2*x)  *(1 - 2*x)*(-1 + 2*x)
------------------------------------------------------------------------------------------------------
                                                    _________________                                 
                                     ___________   /               2                                  
                                 8*\/ x*(1 - x) *\/  1 - (-1 + 2*x)
2x(1−x)(2x(1−x)1−(2x−1)2+1)+(1−2x)1−(2x−1)2(2x−1)8x(1−x)1−(2x−1)2\frac{2 \sqrt{x \left(1 - x\right)} \left(2 \sqrt{x \left(1 - x\right)} \sqrt{1 - \left(2 x - 1\right)^{2}} + 1\right) + \left(1 - 2 x\right) \sqrt{1 - \left(2 x - 1\right)^{2}} \left(2 x - 1\right)}{8 \sqrt{x \left(1 - x\right)} \sqrt{1 - \left(2 x - 1\right)^{2}}} 
(2*sqrt(x*(1 - x))*(1 + 2*sqrt(x*(1 - x))*sqrt(1 - (-1 + 2*x)^2)) + sqrt(1 - (-1 + 2*x)^2)*(1 - 2*x)*(-1 + 2*x))/(8*sqrt(x*(1 - x))*sqrt(1 - (-1 + 2*x)^2))
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