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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
                  /  ___\
   sin(x)      sin\\/ x /
------------ - ----------
    ________        ___  
2*\/ cos(x)     2*\/ x
sin⁡(x)2cos⁡(x)−sin⁡(x)2x\frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} - \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}} 
sin(x)/((2*sqrt(cos(x)))) - sin(sqrt(x))/(2*sqrt(x))
Respuesta numérica [src] 
0.5*cos(x)^(-0.5)*sin(x) - 0.5*x^(-0.5)*sin(sqrt(x))
0.5*cos(x)^(-0.5)*sin(x) - 0.5*x^(-0.5)*sin(sqrt(x))
Unión de expresiones racionales [src] 
  ___            ________    /  ___\
\/ x *sin(x) - \/ cos(x) *sin\\/ x /
------------------------------------
             ___   ________         
         2*\/ x *\/ cos(x)
xsin⁡(x)−sin⁡(x)cos⁡(x)2xcos⁡(x)\frac{\sqrt{x} \sin{\left(x \right)} - \sin{\left(\sqrt{x} \right)} \sqrt{\cos{\left(x \right)}}}{2 \sqrt{x} \sqrt{\cos{\left(x \right)}}} 
(sqrt(x)*sin(x) - sqrt(cos(x))*sin(sqrt(x)))/(2*sqrt(x)*sqrt(cos(x)))
Potencias [src] 
                            /        ___        ___\
      /   -I*x    I*x\      |   -I*\/ x     I*\/ x |
    I*\- e     + e   /    I*\- e         + e       /
- --------------------- + --------------------------
         ______________                ___          
        /  I*x    -I*x             4*\/ x           
       /  e      e                                  
  4*  /   ---- + -----                              
    \/     2       2
−i(eix−e−ix)4eix2+e−ix2+i(eix−e−ix)4x- \frac{i \left(e^{i x} - e^{- i x}\right)}{4 \sqrt{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}} + \frac{i \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)}{4 \sqrt{x}} 
-i*(-exp(-i*x) + exp(i*x))/(4*sqrt(exp(i*x)/2 + exp(-i*x)/2)) + i*(-exp(-i*sqrt(x)) + exp(i*sqrt(x)))/(4*sqrt(x))
Denominador común [src] 
 /  ________    /  ___\     ___       \ 
-\\/ cos(x) *sin\\/ x / - \/ x *sin(x)/ 
----------------------------------------
               ___   ________           
           2*\/ x *\/ cos(x)
−−xsin⁡(x)+sin⁡(x)cos⁡(x)2xcos⁡(x)- \frac{- \sqrt{x} \sin{\left(x \right)} + \sin{\left(\sqrt{x} \right)} \sqrt{\cos{\left(x \right)}}}{2 \sqrt{x} \sqrt{\cos{\left(x \right)}}} 
-(sqrt(cos(x))*sin(sqrt(x)) - sqrt(x)*sin(x))/(2*sqrt(x)*sqrt(cos(x)))
Denominador racional [src] 
      ___           /  ___\         ________       
- 2*\/ x *cos(x)*sin\\/ x / + 2*x*\/ cos(x) *sin(x)
---------------------------------------------------
                     4*x*cos(x)
−2xsin⁡(x)cos⁡(x)+2xsin⁡(x)cos⁡(x)4xcos⁡(x)\frac{- 2 \sqrt{x} \sin{\left(\sqrt{x} \right)} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} \sqrt{\cos{\left(x \right)}}}{4 x \cos{\left(x \right)}} 
(-2*sqrt(x)*cos(x)*sin(sqrt(x)) + 2*x*sqrt(cos(x))*sin(x))/(4*x*cos(x))
Combinatoria [src] 
 /  ________    /  ___\     ___       \ 
-\\/ cos(x) *sin\\/ x / - \/ x *sin(x)/ 
----------------------------------------
               ___   ________           
           2*\/ x *\/ cos(x)
−−xsin⁡(x)+sin⁡(x)cos⁡(x)2xcos⁡(x)- \frac{- \sqrt{x} \sin{\left(x \right)} + \sin{\left(\sqrt{x} \right)} \sqrt{\cos{\left(x \right)}}}{2 \sqrt{x} \sqrt{\cos{\left(x \right)}}} 
-(sqrt(cos(x))*sin(sqrt(x)) - sqrt(x)*sin(x))/(2*sqrt(x)*sqrt(cos(x)))
Parte trigonométrica [src] 
   /    pi\       /  ___   pi\
cos|x - --|    cos|\/ x  - --|
   \    2 /       \        2 /
------------ - ---------------
    ________           ___    
2*\/ cos(x)        2*\/ x
cos⁡(x−π2)2cos⁡(x)−cos⁡(x−π2)2x\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{\cos{\left(x \right)}}} - \frac{\cos{\left(\sqrt{x} - \frac{\pi}{2} \right)}}{2 \sqrt{x}} 
            1                           1           
-------------------------- - -----------------------
      ________                   ___    /  ___   pi\
     /   1        /    pi\   2*\/ x *sec|\/ x  - --|
2*  /  ------ *sec|x - --|              \        2 /
  \/   sec(x)     \    2 /
121sec⁡(x)sec⁡(x−π2)−12xsec⁡(x−π2)\frac{1}{2 \sqrt{\frac{1}{\sec{\left(x \right)}}} \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{2 \sqrt{x} \sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}} 
                         /  ___\
       sin(x)         sin\\/ x /
------------------- - ----------
      _____________        ___  
     /    /    pi\     2*\/ x   
2*  /  sin|x + --|              
  \/      \    2 /
sin⁡(x)2sin⁡(x+π2)−sin⁡(x)2x\frac{\sin{\left(x \right)}}{2 \sqrt{\sin{\left(x + \frac{\pi}{2} \right)}}} - \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}} 
                                                /  ___\      
                  /x\                           |\/ x |      
               tan|-|                        tan|-----|      
                  \2/                           \  2  /      
----------------------------------- - -----------------------
        _____________                       /        /  ___\\
       /        2/x\                    ___ |       2|\/ x ||
      /  1 - tan |-|                  \/ x *|1 + tan |-----||
     /           \2/  /       2/x\\         \        \  2  //
    /    ----------- *|1 + tan |-||                          
   /            2/x\  \        \2//                          
  /      1 + tan |-|                                         
\/               \2/
tan⁡(x2)1−tan⁡2(x2)tan⁡2(x2)+1(tan⁡2(x2)+1)−tan⁡(x2)x(tan⁡2(x2)+1)\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\tan{\left(\frac{\sqrt{x}}{2} \right)}}{\sqrt{x} \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} 
                                                 /  ___\      
                  /x\                            |\/ x |      
               cot|-|                         cot|-----|      
                  \2/                            \  2  /      
------------------------------------ - -----------------------
        ______________                       /        /  ___\\
       /         2/x\                    ___ |       2|\/ x ||
      /  -1 + cot |-|                  \/ x *|1 + cot |-----||
     /            \2/  /       2/x\\         \        \  2  //
    /    ------------ *|1 + cot |-||                          
   /            2/x\   \        \2//                          
  /      1 + cot |-|                                          
\/               \2/
cot⁡(x2)cot⁡2(x2)−1cot⁡2(x2)+1(cot⁡2(x2)+1)−cot⁡(x2)x(cot⁡2(x2)+1)\frac{\cot{\left(\frac{x}{2} \right)}}{\sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\cot{\left(\frac{\sqrt{x}}{2} \right)}}{\sqrt{x} \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} 
             1                         1         
---------------------------- - ------------------
        _____________              ___    /  ___\
       /      1                2*\/ x *csc\\/ x /
2*    /  ----------- *csc(x)                     
     /      /pi    \                             
    /    csc|-- - x|                             
  \/        \2     /
121csc⁡(−x+π2)csc⁡(x)−12xcsc⁡(x)\frac{1}{2 \sqrt{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \csc{\left(x \right)}} - \frac{1}{2 \sqrt{x} \csc{\left(\sqrt{x} \right)}} 
          1                     1         
--------------------- - ------------------
      ________              ___    /  ___\
     /   1              2*\/ x *csc\\/ x /
2*  /  ------ *csc(x)                     
  \/   sec(x)
121sec⁡(x)csc⁡(x)−12xcsc⁡(x)\frac{1}{2 \sqrt{\frac{1}{\sec{\left(x \right)}}} \csc{\left(x \right)}} - \frac{1}{2 \sqrt{x} \csc{\left(\sqrt{x} \right)}} 
1/(2*sqrt(1/sec(x))*csc(x)) - 1/(2*sqrt(x)*csc(sqrt(x)))
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