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Simplificación de expresiones/ Descomposición en fracciones simples/ asin(x/2)-(4-x^2)^(5/2)/(12*x^3)+(4-x^2)^(3/2)/(6*x)+(x*sqrt(4-x^2))/4

¿Cómo vas a descomponer esta asin(x/2)-(4-x^2)^(5/2)/(12*x^3)+(4-x^2)^(3/2)/(6*x)+(x*sqrt(4-x^2))/4 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                  5/2           3/2        ________
          /     2\      /     2\          /      2 
    /x\   \4 - x /      \4 - x /      x*\/  4 - x  
asin|-| - ----------- + ----------- + -------------
    \2/          3          6*x             4      
             12*x
x4−x24+((asin⁡(x2)−(4−x2)5212x3)+(4−x2)326x)\frac{x \sqrt{4 - x^{2}}}{4} + \left(\left(\operatorname{asin}{\left(\frac{x}{2} \right)} - \frac{\left(4 - x^{2}\right)^{\frac{5}{2}}}{12 x^{3}}\right) + \frac{\left(4 - x^{2}\right)^{\frac{3}{2}}}{6 x}\right) 
asin(x/2) - (4 - x^2)^(5/2)/(12*x^3) + (4 - x^2)^(3/2)/((6*x)) + (x*sqrt(4 - x^2))/4
Simplificación general [src] 
       ________        ________          
      /      2        /      2           
  4*\/  4 - x     4*\/  4 - x         /x\
- ------------- + ------------- + asin|-|
          3            3*x            \2/
       3*x
asin⁡(x2)+44−x23x−44−x23x3\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{4 \sqrt{4 - x^{2}}}{3 x} - \frac{4 \sqrt{4 - x^{2}}}{3 x^{3}} 
-4*sqrt(4 - x^2)/(3*x^3) + 4*sqrt(4 - x^2)/(3*x) + asin(x/2)
Respuesta numérica [src] 
0.5*x*(1 - 0.25*x^2)^0.5 + 1.33333333333333*(1 - 0.25*x^2)^1.5/x - 2.66666666666667*(1 - 0.25*x^2)^2.5/x^3 + asin(x/2)
0.5*x*(1 - 0.25*x^2)^0.5 + 1.33333333333333*(1 - 0.25*x^2)^1.5/x - 2.66666666666667*(1 - 0.25*x^2)^2.5/x^3 + asin(x/2)
Denominador común [src] 
       ________           ________          
      /      2       2   /      2           
- 4*\/  4 - x   + 4*x *\/  4 - x         /x\
---------------------------------- + asin|-|
                  3                      \2/
               3*x
asin⁡(x2)+4x24−x2−44−x23x3\operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{4 x^{2} \sqrt{4 - x^{2}} - 4 \sqrt{4 - x^{2}}}{3 x^{3}} 
(-4*sqrt(4 - x^2) + 4*x^2*sqrt(4 - x^2))/(3*x^3) + asin(x/2)
Potencias [src] 
          5/2        ________           3/2          
  /     2\          /      2    /     2\             
  \4 - x /      x*\/  4 - x     \4 - x /          /x\
- ----------- + ------------- + ----------- + asin|-|
         3            4             6*x           \2/
     12*x
x4−x24+asin⁡(x2)+(4−x2)326x−(4−x2)5212x3\frac{x \sqrt{4 - x^{2}}}{4} + \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\left(4 - x^{2}\right)^{\frac{3}{2}}}{6 x} - \frac{\left(4 - x^{2}\right)^{\frac{5}{2}}}{12 x^{3}} 
-(4 - x^2)^(5/2)/(12*x^3) + x*sqrt(4 - x^2)/4 + (4 - x^2)^(3/2)/(6*x) + asin(x/2)
Parte trigonométrica [src] 
          5/2        ________           3/2          
  /     2\          /      2    /     2\             
  \4 - x /      x*\/  4 - x     \4 - x /          /x\
- ----------- + ------------- + ----------- + asin|-|
         3            4             6*x           \2/
     12*x
x4−x24+asin⁡(x2)+(4−x2)326x−(4−x2)5212x3\frac{x \sqrt{4 - x^{2}}}{4} + \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\left(4 - x^{2}\right)^{\frac{3}{2}}}{6 x} - \frac{\left(4 - x^{2}\right)^{\frac{5}{2}}}{12 x^{3}} 
-(4 - x^2)^(5/2)/(12*x^3) + x*sqrt(4 - x^2)/4 + (4 - x^2)^(3/2)/(6*x) + asin(x/2)
Compilar la expresión [src] 
          5/2        ________           3/2          
  /     2\          /      2    /     2\             
  \4 - x /      x*\/  4 - x     \4 - x /          /x\
- ----------- + ------------- + ----------- + asin|-|
         3            4             6*x           \2/
     12*x
x4−x24+asin⁡(x2)+(4−x2)326x−(4−x2)5212x3\frac{x \sqrt{4 - x^{2}}}{4} + \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\left(4 - x^{2}\right)^{\frac{3}{2}}}{6 x} - \frac{\left(4 - x^{2}\right)^{\frac{5}{2}}}{12 x^{3}} 
-(4 - x^2)^(5/2)/(12*x^3) + x*sqrt(4 - x^2)/4 + (4 - x^2)^(3/2)/(6*x) + asin(x/2)
Combinatoria [src] 
       ________                          ________
      /      2       3     /x\      2   /      2 
- 4*\/  4 - x   + 3*x *asin|-| + 4*x *\/  4 - x  
                           \2/                   
-------------------------------------------------
                          3                      
                       3*x
3x3asin⁡(x2)+4x24−x2−44−x23x3\frac{3 x^{3} \operatorname{asin}{\left(\frac{x}{2} \right)} + 4 x^{2} \sqrt{4 - x^{2}} - 4 \sqrt{4 - x^{2}}}{3 x^{3}} 
(-4*sqrt(4 - x^2) + 3*x^3*asin(x/2) + 4*x^2*sqrt(4 - x^2))/(3*x^3)
Unión de expresiones racionales [src] 
          5/2                3/2           ________                
  /     2\         2 /     2\         4   /      2        3     /x\
- \4 - x /    + 2*x *\4 - x /    + 3*x *\/  4 - x   + 12*x *asin|-|
                                                                \2/
-------------------------------------------------------------------
                                   3                               
                               12*x
3x44−x2+12x3asin⁡(x2)+2x2(4−x2)32−(4−x2)5212x3\frac{3 x^{4} \sqrt{4 - x^{2}} + 12 x^{3} \operatorname{asin}{\left(\frac{x}{2} \right)} + 2 x^{2} \left(4 - x^{2}\right)^{\frac{3}{2}} - \left(4 - x^{2}\right)^{\frac{5}{2}}}{12 x^{3}} 
(-(4 - x^2)^(5/2) + 2*x^2*(4 - x^2)^(3/2) + 3*x^4*sqrt(4 - x^2) + 12*x^3*asin(x/2))/(12*x^3)
Abrimos la expresión [src] 
     ________           5/2           3/2          
    /      2    /     2\      /     2\             
x*\/  4 - x     \4 - x /      \4 - x /          /x\
------------- - ----------- + ----------- + asin|-|
      4                3          6*x           \2/
                   12*x
x4−x24+asin⁡(x2)+(4−x2)326x−(4−x2)5212x3\frac{x \sqrt{4 - x^{2}}}{4} + \operatorname{asin}{\left(\frac{x}{2} \right)} + \frac{\left(4 - x^{2}\right)^{\frac{3}{2}}}{6 x} - \frac{\left(4 - x^{2}\right)^{\frac{5}{2}}}{12 x^{3}} 
(x*sqrt(4 - x^2))/4 - (4 - x^2)^(5/2)/(12*x^3) + (4 - x^2)^(3/2)/(6*x) + asin(x/2)
Denominador racional [src] 
     /          5/2                \                 3/2            ________
     |  /     2\          3     /x\|       3 /     2\          5   /      2 
24*x*|- \4 - x /    + 12*x *asin|-|| + 48*x *\4 - x /    + 72*x *\/  4 - x  
     \                          \2//                                        
----------------------------------------------------------------------------
                                        4                                   
                                   288*x
72x54−x2+48x3(4−x2)32+24x(12x3asin⁡(x2)−(4−x2)52)288x4\frac{72 x^{5} \sqrt{4 - x^{2}} + 48 x^{3} \left(4 - x^{2}\right)^{\frac{3}{2}} + 24 x \left(12 x^{3} \operatorname{asin}{\left(\frac{x}{2} \right)} - \left(4 - x^{2}\right)^{\frac{5}{2}}\right)}{288 x^{4}} 
(24*x*(-(4 - x^2)^(5/2) + 12*x^3*asin(x/2)) + 48*x^3*(4 - x^2)^(3/2) + 72*x^5*sqrt(4 - x^2))/(288*x^4)
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