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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
       /  ___\    
    cos\\/ x /    
------------------
    ___    /  ___\
2*\/ x *sin\\/ x /
cos⁡(x)2xsin⁡(x)\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \sin{\left(\sqrt{x} \right)}} 
cos(sqrt(x))/(((2*sqrt(x))*sin(sqrt(x))))
Simplificación general [src] 
        1         
------------------
    ___    /  ___\
2*\/ x *tan\\/ x /
12xtan⁡(x)\frac{1}{2 \sqrt{x} \tan{\left(\sqrt{x} \right)}} 
1/(2*sqrt(x)*tan(sqrt(x)))
Respuesta numérica [src] 
0.5*x^(-0.5)*cos(sqrt(x))/sin(sqrt(x))
0.5*x^(-0.5)*cos(sqrt(x))/sin(sqrt(x))
Denominador racional [src] 
  ___    /  ___\
\/ x *cos\\/ x /
----------------
        /  ___\ 
 2*x*sin\\/ x /
xcos⁡(x)2xsin⁡(x)\frac{\sqrt{x} \cos{\left(\sqrt{x} \right)}}{2 x \sin{\left(\sqrt{x} \right)}} 
sqrt(x)*cos(sqrt(x))/(2*x*sin(sqrt(x)))
Parte trigonométrica [src] 
        1         
------------------
    ___    /  ___\
2*\/ x *tan\\/ x /
12xtan⁡(x)\frac{1}{2 \sqrt{x} \tan{\left(\sqrt{x} \right)}} 
       /  ___\    
    csc\\/ x /    
------------------
    ___    /  ___\
2*\/ x *sec\\/ x /
csc⁡(x)2xsec⁡(x)\frac{\csc{\left(\sqrt{x} \right)}}{2 \sqrt{x} \sec{\left(\sqrt{x} \right)}} 
         /  ___\  
        2|\/ x |  
 1 - tan |-----|  
         \  2  /  
------------------
           /  ___\
    ___    |\/ x |
4*\/ x *tan|-----|
           \  2  /
1−tan⁡2(x2)4xtan⁡(x2)\frac{1 - \tan^{2}{\left(\frac{\sqrt{x}}{2} \right)}}{4 \sqrt{x} \tan{\left(\frac{\sqrt{x}}{2} \right)}} 
       /    ___\   
    sin\2*\/ x /   
-------------------
    ___    2/  ___\
4*\/ x *sin \\/ x /
sin⁡(2x)4xsin⁡2(x)\frac{\sin{\left(2 \sqrt{x} \right)}}{4 \sqrt{x} \sin^{2}{\left(\sqrt{x} \right)}} 
                      /  ___\
                     2|\/ x |
     /  ___\    2*sin |-----|
  sin\\/ x /          \  2  /
------------- - -------------
      /  ___\        /  ___\ 
     2|\/ x |     sin\\/ x / 
2*sin |-----|                
      \  2  /                
-----------------------------
               ___           
           4*\/ x
−2sin⁡2(x2)sin⁡(x)+sin⁡(x)2sin⁡2(x2)4x\frac{- \frac{2 \sin^{2}{\left(\frac{\sqrt{x}}{2} \right)}}{\sin{\left(\sqrt{x} \right)}} + \frac{\sin{\left(\sqrt{x} \right)}}{2 \sin^{2}{\left(\frac{\sqrt{x}}{2} \right)}}}{4 \sqrt{x}} 
      /  ___\        /  ___     \
      |\/ x |        |\/ x    pi|
   cos|-----|     cos|----- - --|
      \  2  /        \  2     2 /
--------------- - ---------------
   /  ___     \         /  ___\  
   |\/ x    pi|         |\/ x |  
cos|----- - --|      cos|-----|  
   \  2     2 /         \  2  /  
---------------------------------
                 ___             
             4*\/ x
cos⁡(x2)cos⁡(x2−π2)−cos⁡(x2−π2)cos⁡(x2)4x\frac{\frac{\cos{\left(\frac{\sqrt{x}}{2} \right)}}{\cos{\left(\frac{\sqrt{x}}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{\sqrt{x}}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{\sqrt{x}}{2} \right)}}}{4 \sqrt{x}} 
                  /  ___\
      1           |\/ x |
- ---------- + cot|-----|
     /  ___\      \  2  /
     |\/ x |             
  cot|-----|             
     \  2  /             
-------------------------
             ___         
         4*\/ x
cot⁡(x2)−1cot⁡(x2)4x\frac{\cot{\left(\frac{\sqrt{x}}{2} \right)} - \frac{1}{\cot{\left(\frac{\sqrt{x}}{2} \right)}}}{4 \sqrt{x}} 
   /  ___\
cot\\/ x /
----------
     ___  
 2*\/ x
cot⁡(x)2x\frac{\cot{\left(\sqrt{x} \right)}}{2 \sqrt{x}} 
      /  ___\        /       ___\
      |\/ x |        |pi   \/ x |
   csc|-----|     csc|-- - -----|
      \  2  /        \2      2  /
--------------- - ---------------
   /       ___\         /  ___\  
   |pi   \/ x |         |\/ x |  
csc|-- - -----|      csc|-----|  
   \2      2  /         \  2  /  
---------------------------------
                 ___             
             4*\/ x
csc⁡(x2)csc⁡(−x2+π2)−csc⁡(−x2+π2)csc⁡(x2)4x\frac{\frac{\csc{\left(\frac{\sqrt{x}}{2} \right)}}{\csc{\left(- \frac{\sqrt{x}}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- \frac{\sqrt{x}}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{\sqrt{x}}{2} \right)}}}{4 \sqrt{x}} 
                       /  ___\
                      2|\/ x |
     /  ___\     2*sin |-----|
  sin\\/ x /           \  2  /
-------------- - -------------
       /  ___\        /  ___\ 
1 - cos\\/ x /     sin\\/ x / 
------------------------------
               ___            
           4*\/ x
−2sin⁡2(x2)sin⁡(x)+sin⁡(x)1−cos⁡(x)4x\frac{- \frac{2 \sin^{2}{\left(\frac{\sqrt{x}}{2} \right)}}{\sin{\left(\sqrt{x} \right)}} + \frac{\sin{\left(\sqrt{x} \right)}}{1 - \cos{\left(\sqrt{x} \right)}}}{4 \sqrt{x}} 
          /  ___\ 
         2|\/ x | 
 -1 + cot |-----| 
          \  2  / 
------------------
           /  ___\
    ___    |\/ x |
4*\/ x *cot|-----|
           \  2  /
cot⁡2(x2)−14xcot⁡(x2)\frac{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} - 1}{4 \sqrt{x} \cot{\left(\frac{\sqrt{x}}{2} \right)}} 
                /  ___\
    1           |\/ x |
---------- - tan|-----|
   /  ___\      \  2  /
   |\/ x |             
tan|-----|             
   \  2  /             
-----------------------
            ___        
        4*\/ x
−tan⁡(x2)+1tan⁡(x2)4x\frac{- \tan{\left(\frac{\sqrt{x}}{2} \right)} + \frac{1}{\tan{\left(\frac{\sqrt{x}}{2} \right)}}}{4 \sqrt{x}} 
   /  ___     \         /  ___\  
   |\/ x    pi|         |\/ x |  
sec|----- - --|      sec|-----|  
   \  2     2 /         \  2  /  
--------------- - ---------------
      /  ___\        /  ___     \
      |\/ x |        |\/ x    pi|
   sec|-----|     sec|----- - --|
      \  2  /        \  2     2 /
---------------------------------
                 ___             
             4*\/ x
−sec⁡(x2)sec⁡(x2−π2)+sec⁡(x2−π2)sec⁡(x2)4x\frac{- \frac{\sec{\left(\frac{\sqrt{x}}{2} \right)}}{\sec{\left(\frac{\sqrt{x}}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{\sqrt{x}}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{\sqrt{x}}{2} \right)}}}{4 \sqrt{x}} 
    /  ___   pi\  
 sec|\/ x  - --|  
    \        2 /  
------------------
    ___    /  ___\
2*\/ x *sec\\/ x /
sec⁡(x−π2)2xsec⁡(x)\frac{\sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}}{2 \sqrt{x} \sec{\left(\sqrt{x} \right)}} 
     /  ___\      /  ___\
     |\/ x |      |\/ x |
- tan|-----| + cot|-----|
     \  2  /      \  2  /
-------------------------
             ___         
         4*\/ x
−tan⁡(x2)+cot⁡(x2)4x\frac{- \tan{\left(\frac{\sqrt{x}}{2} \right)} + \cot{\left(\frac{\sqrt{x}}{2} \right)}}{4 \sqrt{x}} 
          /  ___\      
       cos\\/ x /      
-----------------------
    ___    /  ___   pi\
2*\/ x *cos|\/ x  - --|
           \        2 /
cos⁡(x)2xcos⁡(x−π2)\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \cos{\left(\sqrt{x} - \frac{\pi}{2} \right)}} 
    /  ___   pi\  
 sin|\/ x  + --|  
    \        2 /  
------------------
    ___    /  ___\
2*\/ x *sin\\/ x /
sin⁡(x+π2)2xsin⁡(x)\frac{\sin{\left(\sqrt{x} + \frac{\pi}{2} \right)}}{2 \sqrt{x} \sin{\left(\sqrt{x} \right)}} 
          /  ___\      
       csc\\/ x /      
-----------------------
    ___    /pi     ___\
2*\/ x *csc|-- - \/ x |
           \2         /
csc⁡(x)2xcsc⁡(−x+π2)\frac{\csc{\left(\sqrt{x} \right)}}{2 \sqrt{x} \csc{\left(- \sqrt{x} + \frac{\pi}{2} \right)}} 
csc(sqrt(x))/(2*sqrt(x)*csc(pi/2 - sqrt(x)))
Potencias [src] 
     /     ___         ___\   
     | I*\/ x     -I*\/ x |   
     |e          e        |   
   I*|-------- + ---------|   
     \   2           2    /   
------------------------------
      /        ___        ___\
  ___ |   -I*\/ x     I*\/ x |
\/ x *\- e         + e       /
i(eix2+e−ix2)x(eix−e−ix)\frac{i \left(\frac{e^{i \sqrt{x}}}{2} + \frac{e^{- i \sqrt{x}}}{2}\right)}{\sqrt{x} \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)} 
i*(exp(i*sqrt(x))/2 + exp(-i*sqrt(x))/2)/(sqrt(x)*(-exp(-i*sqrt(x)) + exp(i*sqrt(x))))
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