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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
                              /     ___             \
asin(x)        1          ___ |   \/ x       asin(x)|
------- - ----------- + \/ x *|----------- + -------|
   2          _______         |   ________       ___|
          2*\/ 1 - x          |  /      2    2*\/ x |
                              \\/  1 - x            /
$$\sqrt{x} \left(\frac{\sqrt{x}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{2 \sqrt{x}}\right) + \left(\frac{\operatorname{asin}{\left(x \right)}}{2} - \frac{1}{2 \sqrt{1 - x}}\right)$$ 
asin(x)/2 - 1/(2*sqrt(1 - x)) + sqrt(x)*(sqrt(x)/sqrt(1 - x^2) + asin(x)/((2*sqrt(x))))
Simplificación general [src] 
       1             x               
- ----------- + ----------- + asin(x)
      _______      ________          
  2*\/ 1 - x      /      2           
                \/  1 - x
$$\frac{x}{\sqrt{1 - x^{2}}} + \operatorname{asin}{\left(x \right)} - \frac{1}{2 \sqrt{1 - x}}$$ 
-1/(2*sqrt(1 - x)) + x/sqrt(1 - x^2) + asin(x)
Abrimos la expresión [src] 
                      /     ___             \          
       1          ___ |   \/ x       asin(x)|   asin(x)
- ----------- + \/ x *|----------- + -------| + -------
      _______         |   ________       ___|      2   
  2*\/ 1 - x          |  /      2    2*\/ x |          
                      \\/  1 - x            /
$$\sqrt{x} \left(\frac{\sqrt{x}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{2 \sqrt{x}}\right) + \frac{\operatorname{asin}{\left(x \right)}}{2} - \frac{1}{2 \sqrt{1 - x}}$$ 
-1/(2*sqrt(1 - x)) + sqrt(x)*(sqrt(x)/sqrt(1 - x^2) + asin(x)/(2*sqrt(x))) + asin(x)/2
Denominador racional [src] 
                             ________                                                                                       ________                 
     3/2              5/2   /      2       5/2               ___   _______      5/2   _______       ___              3/2   /      2       7/2        
- 8*x   *asin(x) - 8*x   *\/  1 - x   - 8*x   *asin(x) - 4*\/ x *\/ 1 - x  + 4*x   *\/ 1 - x  + 8*\/ x *asin(x) + 8*x   *\/  1 - x   + 8*x   *asin(x)
-----------------------------------------------------------------------------------------------------------------------------------------------------
                                                                  ___          /      2\                                                             
                                                              8*\/ x *(-1 + x)*\-1 + x /
$$\frac{8 x^{\frac{7}{2}} \operatorname{asin}{\left(x \right)} + 4 x^{\frac{5}{2}} \sqrt{1 - x} - 8 x^{\frac{5}{2}} \sqrt{1 - x^{2}} - 8 x^{\frac{5}{2}} \operatorname{asin}{\left(x \right)} + 8 x^{\frac{3}{2}} \sqrt{1 - x^{2}} - 8 x^{\frac{3}{2}} \operatorname{asin}{\left(x \right)} - 4 \sqrt{x} \sqrt{1 - x} + 8 \sqrt{x} \operatorname{asin}{\left(x \right)}}{8 \sqrt{x} \left(x - 1\right) \left(x^{2} - 1\right)}$$ 
(-8*x^(3/2)*asin(x) - 8*x^(5/2)*sqrt(1 - x^2) - 8*x^(5/2)*asin(x) - 4*sqrt(x)*sqrt(1 - x) + 4*x^(5/2)*sqrt(1 - x) + 8*sqrt(x)*asin(x) + 8*x^(3/2)*sqrt(1 - x^2) + 8*x^(7/2)*asin(x))/(8*sqrt(x)*(-1 + x)*(-1 + x^2))
Respuesta numérica [src] 
0.5*asin(x) - 0.5*(1.0 - x)^(-0.5) + x^0.5*(x^0.5*(1.0 - x^2)^(-0.5) + 0.5*x^(-0.5)*asin(x))
0.5*asin(x) - 0.5*(1.0 - x)^(-0.5) + x^0.5*(x^0.5*(1.0 - x^2)^(-0.5) + 0.5*x^(-0.5)*asin(x))
Compilar la expresión [src] 
                              /     ___             \
asin(x)        1          ___ |   \/ x       asin(x)|
------- - ----------- + \/ x *|----------- + -------|
   2          _______         |   ________       ___|
          2*\/ 1 - x          |  /      2    2*\/ x |
                              \\/  1 - x            /
$$\sqrt{x} \left(\frac{\sqrt{x}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{2 \sqrt{x}}\right) + \frac{\operatorname{asin}{\left(x \right)}}{2} - \frac{1}{2 \sqrt{1 - x}}$$ 
asin(x)/2 - 1/(2*sqrt(1 - x)) + sqrt(x)*(sqrt(x)/sqrt(1 - x^2) + asin(x)/(2*sqrt(x)))
Potencias [src] 
                              /     ___             \
asin(x)        1          ___ |   \/ x       asin(x)|
------- - ----------- + \/ x *|----------- + -------|
   2          _______         |   ________       ___|
          2*\/ 1 - x          |  /      2    2*\/ x |
                              \\/  1 - x            /
$$\sqrt{x} \left(\frac{\sqrt{x}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{2 \sqrt{x}}\right) + \frac{\operatorname{asin}{\left(x \right)}}{2} - \frac{1}{2 \sqrt{1 - x}}$$ 
asin(x)/2 - 1/(2*sqrt(1 - x)) + sqrt(x)*(sqrt(x)/sqrt(1 - x^2) + asin(x)/(2*sqrt(x)))
Unión de expresiones racionales [src] 
          /         ________        \      ________                         
  _______ |        /      2         |     /      2  /       _______        \
\/ 1 - x *\2*x + \/  1 - x  *asin(x)/ + \/  1 - x  *\-1 + \/ 1 - x *asin(x)/
----------------------------------------------------------------------------
                                         ________                           
                              _______   /      2                            
                          2*\/ 1 - x *\/  1 - x
$$\frac{\sqrt{1 - x} \left(2 x + \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}\right) + \sqrt{1 - x^{2}} \left(\sqrt{1 - x} \operatorname{asin}{\left(x \right)} - 1\right)}{2 \sqrt{1 - x} \sqrt{1 - x^{2}}}$$ 
(sqrt(1 - x)*(2*x + sqrt(1 - x^2)*asin(x)) + sqrt(1 - x^2)*(-1 + sqrt(1 - x)*asin(x)))/(2*sqrt(1 - x)*sqrt(1 - x^2))
Combinatoria [src] 
     ________                                  ________        
    /      2          _______       _______   /      2         
- \/  1 - x   + 2*x*\/ 1 - x  + 2*\/ 1 - x *\/  1 - x  *asin(x)
---------------------------------------------------------------
                   ___________________   _______               
               2*\/ -(1 + x)*(-1 + x) *\/ 1 - x
$$\frac{2 x \sqrt{1 - x} + 2 \sqrt{1 - x} \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)} - \sqrt{1 - x^{2}}}{2 \sqrt{- \left(x - 1\right) \left(x + 1\right)} \sqrt{1 - x}}$$ 
(-sqrt(1 - x^2) + 2*x*sqrt(1 - x) + 2*sqrt(1 - x)*sqrt(1 - x^2)*asin(x))/(2*sqrt(-(1 + x)*(-1 + x))*sqrt(1 - x))
Parte trigonométrica [src] 
                              /     ___             \
asin(x)        1          ___ |   \/ x       asin(x)|
------- - ----------- + \/ x *|----------- + -------|
   2          _______         |   ________       ___|
          2*\/ 1 - x          |  /      2    2*\/ x |
                              \\/  1 - x            /
$$\sqrt{x} \left(\frac{\sqrt{x}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{2 \sqrt{x}}\right) + \frac{\operatorname{asin}{\left(x \right)}}{2} - \frac{1}{2 \sqrt{1 - x}}$$ 
asin(x)/2 - 1/(2*sqrt(1 - x)) + sqrt(x)*(sqrt(x)/sqrt(1 - x^2) + asin(x)/(2*sqrt(x)))
Denominador común [src] 
     ________                          
    /      2          _______          
- \/  1 - x   + 2*x*\/ 1 - x           
----------------------------- + asin(x)
                  ________             
       _______   /      2              
   2*\/ 1 - x *\/  1 - x
$$\operatorname{asin}{\left(x \right)} + \frac{2 x \sqrt{1 - x} - \sqrt{1 - x^{2}}}{2 \sqrt{1 - x} \sqrt{1 - x^{2}}}$$ 
(-sqrt(1 - x^2) + 2*x*sqrt(1 - x))/(2*sqrt(1 - x)*sqrt(1 - x^2)) + asin(x)
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