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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
           sin(x)           
----------------------------
        ____________________
       /             /   2\ 
      /              |/2\ | 
     /               ||-| | 
    /                \\a/ / 
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡(2a)2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\left(\frac{2}{a}\right)^{2}}{\left(x \right)}}} 
sin(x)/((a*sqrt(1 - cos(x)^((2/a)^2))))
Simplificación general [src] 
         sin(x)         
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡4a2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
Potencias [src] 
           /   -I*x    I*x\        
        -I*\- e     + e   /        
-----------------------------------
             ______________________
            /                   4  
           /                    -- 
          /                      2 
         /                      a  
        /         / I*x    -I*x\   
       /          |e      e    |   
2*a*  /       1 - |---- + -----|   
    \/            \ 2       2  /
−i(eix−e−ix)2a1−(eix2+e−ix2)4a2- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 a \sqrt{1 - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\frac{4}{a^{2}}}}} 
         sin(x)         
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡4a2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
         sin(x)         
------------------------
         _______________
        /            1  
       /             -- 
      /               2 
     /               a  
    /            4      
a*\/      1 - cos (x)
sin⁡(x)a1−(cos⁡4(x))1a2\frac{\sin{\left(x \right)}}{a \sqrt{1 - \left(\cos^{4}{\left(x \right)}\right)^{\frac{1}{a^{2}}}}} 
sin(x)/(a*sqrt(1 - (cos(x)^4)^(a^(-2))))
Unión de expresiones racionales [src] 
         sin(x)         
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡4a2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
Combinatoria [src] 
                            sin(x)                            
--------------------------------------------------------------
        ______________________________________________________
       /  /            1 \ /            2 \ /             1 \ 
      /   |            --| |            --| |             --| 
     /    |             2| |             2| |              2| 
    /     |            a | |            a | |             a | 
a*\/     -\1 + (cos(x))  /*\1 + (cos(x))  /*\-1 + (cos(x))  /
sin⁡(x)a−(cos⁡1a2(x)−1)(cos⁡1a2(x)+1)(cos⁡2a2(x)+1)\frac{\sin{\left(x \right)}}{a \sqrt{- \left(\cos^{\frac{1}{a^{2}}}{\left(x \right)} - 1\right) \left(\cos^{\frac{1}{a^{2}}}{\left(x \right)} + 1\right) \left(\cos^{\frac{2}{a^{2}}}{\left(x \right)} + 1\right)}} 
sin(x)/(a*sqrt(-(1 + cos(x)^(a^(-2)))*(1 + cos(x)^(2/a^2))*(-1 + cos(x)^(a^(-2)))))
Denominador común [src] 
         sin(x)         
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡4a2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
Denominador racional [src] 
       ________________        
      /             4          
     /              --         
    /                2         
   /                a          
-\/     1 - (cos(x))   *sin(x) 
-------------------------------
        /             4 \      
        |             --|      
        |              2|      
        |             a |      
      a*\-1 + (cos(x))  /
−1−cos⁡4a2(x)sin⁡(x)a(cos⁡4a2(x)−1)- \frac{\sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}} \sin{\left(x \right)}}{a \left(\cos^{\frac{4}{a^{2}}}{\left(x \right)} - 1\right)} 
-sqrt(1 - cos(x)^(4/a^2))*sin(x)/(a*(-1 + cos(x)^(4/a^2)))
Parte trigonométrica [src] 
         /    pi\       
      cos|x - --|       
         \    2 /       
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
cos⁡(x−π2)a1−cos⁡4a2(x)\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
                   1                    
----------------------------------------
            _____________________       
           /                  4         
          /                   --        
         /                     2        
        /                     a         
       /         /     1     \          
a*    /      1 - |-----------|   *csc(x)
     /           |   /pi    \|          
    /            |csc|-- - x||          
  \/             \   \2     //
1a1−(1csc⁡(−x+π2))4a2csc⁡(x)\frac{1}{a \sqrt{1 - \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{4}{a^{2}}}} \csc{\left(x \right)}} 
                  1                   
--------------------------------------
          ________________            
         /             4              
        /              --             
       /                2             
      /                a              
     /         /  1   \       /    pi\
a*  /      1 - |------|   *sec|x - --|
  \/           \sec(x)/       \    2 /
1a1−(1sec⁡(x))4a2sec⁡(x−π2)\frac{1}{a \sqrt{1 - \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{4}{a^{2}}}} \sec{\left(x - \frac{\pi}{2} \right)}} 
             sin(x)            
-------------------------------
          _____________________
         /                  4  
        /                   -- 
       /                     2 
      /                     a  
     /         /   /    pi\\   
a*  /      1 - |sin|x + --||   
  \/           \   \    2 //
sin⁡(x)a1−sin⁡4a2(x+π2)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \sin^{\frac{4}{a^{2}}}{\left(x + \frac{\pi}{2} \right)}}} 
                          /x\                     
                     2*cot|-|                     
                          \2/                     
--------------------------------------------------
                            ______________________
                           /                   4  
                          /                    -- 
                         /                      2 
                        /                      a  
                       /         /        2/x\\   
                      /          |-1 + cot |-||   
  /       2/x\\      /           |         \2/|   
a*|1 + cot |-||*    /        1 - |------------|   
  \        \2//    /             |       2/x\ |   
                  /              |1 + cot |-| |   
                \/               \        \2/ /
2cot⁡(x2)a1−(cot⁡2(x2)−1cot⁡2(x2)+1)4a2(cot⁡2(x2)+1)\frac{2 \cot{\left(\frac{x}{2} \right)}}{a \sqrt{1 - \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{4}{a^{2}}}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                1                
---------------------------------
          ________________       
         /             4         
        /              --        
       /                2        
      /                a         
     /         /  1   \          
a*  /      1 - |------|   *csc(x)
  \/           \sec(x)/
1a1−(1sec⁡(x))4a2csc⁡(x)\frac{1}{a \sqrt{1 - \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{4}{a^{2}}}} \csc{\left(x \right)}} 
         sin(x)         
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡4a2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
                          /x\                    
                     2*tan|-|                    
                          \2/                    
-------------------------------------------------
                            _____________________
                           /                  4  
                          /                   -- 
                         /                     2 
                        /                     a  
                       /         /       2/x\\   
                      /          |1 - tan |-||   
  /       2/x\\      /           |        \2/|   
a*|1 + tan |-||*    /        1 - |-----------|   
  \        \2//    /             |       2/x\|   
                  /              |1 + tan |-||   
                \/               \        \2//
2tan⁡(x2)a1−(1−tan⁡2(x2)tan⁡2(x2)+1)4a2(tan⁡2(x2)+1)\frac{2 \tan{\left(\frac{x}{2} \right)}}{a \sqrt{1 - \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{4}{a^{2}}}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
2*tan(x/2)/(a*(1 + tan(x/2)^2)*sqrt(1 - ((1 - tan(x/2)^2)/(1 + tan(x/2)^2))^(4/a^2)))
Respuesta numérica [src] 
(1.0 - cos(x)^(4.0/a^2))^(-0.5)*sin(x)/a
(1.0 - cos(x)^(4.0/a^2))^(-0.5)*sin(x)/a
Abrimos la expresión [src] 
         sin(x)         
------------------------
        ________________
       /             4  
      /              -- 
     /                2 
    /                a  
a*\/     1 - (cos(x))
sin⁡(x)a1−cos⁡4a2(x)\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}} 
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
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