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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
                              x - 1                    /x   1\
  _______________         1 + -----           (-1 + x)*|- - -|
\/ x*x - 2*x - 1              x - 1                    \2   2/
----------------- - ---------------------- + -----------------
        2                            2         _______________
                              _______        \/ x*x - 2*x - 1 
                    x - 1 + \/ x - 1   - 2
(x−1)(x2−12)(−2x+xx)−1+(−1+x−1x−1((x−1)2+(x−1))−2+(−2x+xx)−12)\frac{\left(x - 1\right) \left(\frac{x}{2} - \frac{1}{2}\right)}{\sqrt{\left(- 2 x + x x\right) - 1}} + \left(- \frac{1 + \frac{x - 1}{x - 1}}{\left(\left(\sqrt{x - 1}\right)^{2} + \left(x - 1\right)\right) - 2} + \frac{\sqrt{\left(- 2 x + x x\right) - 1}}{2}\right) 
sqrt(x*x - 2*x - 1)/2 - (1 + (x - 1)/(x - 1))/(x - 1 + (sqrt(x - 1))^2 - 2) + ((-1 + x)*(x/2 - 1/2))/sqrt(x*x - 2*x - 1)
Simplificación general [src] 
        _______________             
 3     /       2             2      
x  - \/  -1 + x  - 2*x  - 4*x  + 4*x
------------------------------------
                _______________     
               /       2            
    (-2 + x)*\/  -1 + x  - 2*x
x3−4x2+4x−x2−2x−1(x−2)x2−2x−1\frac{x^{3} - 4 x^{2} + 4 x - \sqrt{x^{2} - 2 x - 1}}{\left(x - 2\right) \sqrt{x^{2} - 2 x - 1}} 
(x^3 - sqrt(-1 + x^2 - 2*x) - 4*x^2 + 4*x)/((-2 + x)*sqrt(-1 + x^2 - 2*x))
Respuesta numérica [src] 
0.707106781186548*(-0.5 - x + 0.5*x^2)^0.5 - 2.0/(-3.0 + x + (-1.0 + x)^1.0) + 0.707106781186547*(-0.5 - x + 0.5*x^2)^(-0.5)*(-0.5 + 0.5*x)*(-1.0 + x)
0.707106781186548*(-0.5 - x + 0.5*x^2)^0.5 - 2.0/(-3.0 + x + (-1.0 + x)^1.0) + 0.707106781186547*(-0.5 - x + 0.5*x^2)^(-0.5)*(-0.5 + 0.5*x)*(-1.0 + x)
Combinatoria [src] 
        _______________             
 3     /       2             2      
x  - \/  -1 + x  - 2*x  - 4*x  + 4*x
------------------------------------
                _______________     
               /       2            
    (-2 + x)*\/  -1 + x  - 2*x
x3−4x2+4x−x2−2x−1(x−2)x2−2x−1\frac{x^{3} - 4 x^{2} + 4 x - \sqrt{x^{2} - 2 x - 1}}{\left(x - 2\right) \sqrt{x^{2} - 2 x - 1}} 
(x^3 - sqrt(-1 + x^2 - 2*x) - 4*x^2 + 4*x)/((-2 + x)*sqrt(-1 + x^2 - 2*x))
Potencias [src] 
   _______________                       /  1   x\
  /       2                     (-1 + x)*|- - + -|
\/  -1 + x  - 2*x       2                \  2   2/
------------------ - -------- + ------------------
        2            -4 + 2*x      _______________
                                  /       2       
                                \/  -1 + x  - 2*x
(x2−12)(x−1)x2−2x−1+x2−2x−12−22x−4\frac{\left(\frac{x}{2} - \frac{1}{2}\right) \left(x - 1\right)}{\sqrt{x^{2} - 2 x - 1}} + \frac{\sqrt{x^{2} - 2 x - 1}}{2} - \frac{2}{2 x - 4} 
sqrt(-1 + x^2 - 2*x)/2 - 2/(-4 + 2*x) + (-1 + x)*(-1/2 + x/2)/sqrt(-1 + x^2 - 2*x)
Parte trigonométrica [src] 
   _______________                       /  1   x\
  /       2                     (-1 + x)*|- - + -|
\/  -1 + x  - 2*x       2                \  2   2/
------------------ - -------- + ------------------
        2            -4 + 2*x      _______________
                                  /       2       
                                \/  -1 + x  - 2*x
(x2−12)(x−1)x2−2x−1+x2−2x−12−22x−4\frac{\left(\frac{x}{2} - \frac{1}{2}\right) \left(x - 1\right)}{\sqrt{x^{2} - 2 x - 1}} + \frac{\sqrt{x^{2} - 2 x - 1}}{2} - \frac{2}{2 x - 4} 
sqrt(-1 + x^2 - 2*x)/2 - 2/(-4 + 2*x) + (-1 + x)*(-1/2 + x/2)/sqrt(-1 + x^2 - 2*x)
Denominador común [src] 
             _______________                 
      3     /       2             2          
     x  - \/  -1 + x  - 2*x  - 4*x  + 4*x    
---------------------------------------------
       _______________        _______________
      /       2              /       2       
- 2*\/  -1 + x  - 2*x  + x*\/  -1 + x  - 2*x
x3−4x2+4x−x2−2x−1xx2−2x−1−2x2−2x−1\frac{x^{3} - 4 x^{2} + 4 x - \sqrt{x^{2} - 2 x - 1}}{x \sqrt{x^{2} - 2 x - 1} - 2 \sqrt{x^{2} - 2 x - 1}} 
(x^3 - sqrt(-1 + x^2 - 2*x) - 4*x^2 + 4*x)/(-2*sqrt(-1 + x^2 - 2*x) + x*sqrt(-1 + x^2 - 2*x))
Denominador racional [src] 
                                  _______________            _______________                   _______________                       3/2                       3/2           _______________                     _______________                     _______________
         2              3        /       2              2   /       2                     2   /       2             2 /      2      \           /      2      \             /       2             2         2   /       2                       2   /       2       
16 - 48*x  + 16*x + 16*x  + 16*\/  -1 + x  - 2*x  - 16*x *\/  -1 + x  - 2*x  - 16*(-1 + x) *\/  -1 + x  - 2*x  - 8*x *\-1 + x  - 2*x/    + 24*x*\-1 + x  - 2*x/    + 32*x*\/  -1 + x  - 2*x  - 8*x *(-1 + x) *\/  -1 + x  - 2*x  + 24*x*(-1 + x) *\/  -1 + x  - 2*x 
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                      /     2      \                                                                                                                
                                                                                                                8*(-1 + x)*(-4 + 2*x)*\1 - x  + 2*x/
16x3−8x2(x−1)2x2−2x−1−8x2(x2−2x−1)32−16x2x2−2x−1−48x2+24x(x−1)2x2−2x−1+24x(x2−2x−1)32+32xx2−2x−1+16x−16(x−1)2x2−2x−1+16x2−2x−1+168(x−1)(2x−4)(−x2+2x+1)\frac{16 x^{3} - 8 x^{2} \left(x - 1\right)^{2} \sqrt{x^{2} - 2 x - 1} - 8 x^{2} \left(x^{2} - 2 x - 1\right)^{\frac{3}{2}} - 16 x^{2} \sqrt{x^{2} - 2 x - 1} - 48 x^{2} + 24 x \left(x - 1\right)^{2} \sqrt{x^{2} - 2 x - 1} + 24 x \left(x^{2} - 2 x - 1\right)^{\frac{3}{2}} + 32 x \sqrt{x^{2} - 2 x - 1} + 16 x - 16 \left(x - 1\right)^{2} \sqrt{x^{2} - 2 x - 1} + 16 \sqrt{x^{2} - 2 x - 1} + 16}{8 \left(x - 1\right) \left(2 x - 4\right) \left(- x^{2} + 2 x + 1\right)} 
(16 - 48*x^2 + 16*x + 16*x^3 + 16*sqrt(-1 + x^2 - 2*x) - 16*x^2*sqrt(-1 + x^2 - 2*x) - 16*(-1 + x)^2*sqrt(-1 + x^2 - 2*x) - 8*x^2*(-1 + x^2 - 2*x)^(3/2) + 24*x*(-1 + x^2 - 2*x)^(3/2) + 32*x*sqrt(-1 + x^2 - 2*x) - 8*x^2*(-1 + x)^2*sqrt(-1 + x^2 - 2*x) + 24*x*(-1 + x)^2*sqrt(-1 + x^2 - 2*x))/(8*(-1 + x)*(-4 + 2*x)*(1 - x^2 + 2*x))
Compilar la expresión [src] 
   _______________                       /  1   x\
  /       2                     (-1 + x)*|- - + -|
\/  -1 + x  - 2*x       2                \  2   2/
------------------ - -------- + ------------------
        2            -4 + 2*x      _______________
                                  /       2       
                                \/  -1 + x  - 2*x
(x2−12)(x−1)x2−2x−1+x2−2x−12−22x−4\frac{\left(\frac{x}{2} - \frac{1}{2}\right) \left(x - 1\right)}{\sqrt{x^{2} - 2 x - 1}} + \frac{\sqrt{x^{2} - 2 x - 1}}{2} - \frac{2}{2 x - 4} 
sqrt(-1 + x^2 - 2*x)/2 - 2/(-4 + 2*x) + (-1 + x)*(-1/2 + x/2)/sqrt(-1 + x^2 - 2*x)
Unión de expresiones racionales [src] 
        2              _________________ /       _________________         \
(-1 + x) *(-2 + x) + \/ -1 + x*(-2 + x) *\-2 + \/ -1 + x*(-2 + x) *(-2 + x)/
----------------------------------------------------------------------------
                           _________________                                
                       2*\/ -1 + x*(-2 + x) *(-2 + x)
(x−2)(x−1)2+x(x−2)−1((x−2)x(x−2)−1−2)2(x−2)x(x−2)−1\frac{\left(x - 2\right) \left(x - 1\right)^{2} + \sqrt{x \left(x - 2\right) - 1} \left(\left(x - 2\right) \sqrt{x \left(x - 2\right) - 1} - 2\right)}{2 \left(x - 2\right) \sqrt{x \left(x - 2\right) - 1}} 
((-1 + x)^2*(-2 + x) + sqrt(-1 + x*(-2 + x))*(-2 + sqrt(-1 + x*(-2 + x))*(-2 + x)))/(2*sqrt(-1 + x*(-2 + x))*(-2 + x))
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