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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  ___                
\/ x *cos(x) + sin(x)
---------------------
           ___       
       2*\/ x
xcos⁡(x)+sin⁡(x)2x\frac{\sqrt{x} \cos{\left(x \right)} + \sin{\left(x \right)}}{2 \sqrt{x}} 
(sqrt(x)*cos(x) + sin(x))/((2*sqrt(x)))
Simplificación general [src] 
cos(x)    sin(x)
------ + -------
  2          ___
         2*\/ x
cos⁡(x)2+sin⁡(x)2x\frac{\cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}} 
cos(x)/2 + sin(x)/(2*sqrt(x))
Respuesta numérica [src] 
0.5*x^(-0.5)*(x^0.5*cos(x) + sin(x))
0.5*x^(-0.5)*(x^0.5*cos(x) + sin(x))
Denominador común [src] 
cos(x)    sin(x)
------ + -------
  2          ___
         2*\/ x
cos⁡(x)2+sin⁡(x)2x\frac{\cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}} 
cos(x)/2 + sin(x)/(2*sqrt(x))
Denominador racional [src] 
             ___       
x*cos(x) + \/ x *sin(x)
-----------------------
          2*x
xsin⁡(x)+xcos⁡(x)2x\frac{\sqrt{x} \sin{\left(x \right)} + x \cos{\left(x \right)}}{2 x} 
(x*cos(x) + sqrt(x)*sin(x))/(2*x)
Potencias [src] 
      / I*x    -I*x\     /   -I*x    I*x\
  ___ |e      e    |   I*\- e     + e   /
\/ x *|---- + -----| - ------------------
      \ 2       2  /           2         
-----------------------------------------
                     ___                 
                 2*\/ x
x(eix2+e−ix2)−i(eix−e−ix)22x\frac{\sqrt{x} \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) - \frac{i \left(e^{i x} - e^{- i x}\right)}{2}}{2 \sqrt{x}} 
           ___       
sin(x)   \/ x *cos(x)
------ + ------------
  2           2      
---------------------
          ___        
        \/ x
xcos⁡(x)2+sin⁡(x)2x\frac{\frac{\sqrt{x} \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}}{\sqrt{x}} 
(sin(x)/2 + sqrt(x)*cos(x)/2)/sqrt(x)
Abrimos la expresión [src] 
cos(x)    sin(x)
------ + -------
  2          ___
         2*\/ x
cos⁡(x)2+sin⁡(x)2x\frac{\cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2 \sqrt{x}} 
cos(x)/2 + sin(x)/(2*sqrt(x))
Parte trigonométrica [src] 
       /x\      ___ /        2/x\\
  2*cot|-|    \/ x *|-1 + cot |-||
       \2/          \         \2//
----------- + --------------------
       2/x\              2/x\     
1 + cot |-|       1 + cot |-|     
        \2/               \2/     
----------------------------------
                 ___              
             2*\/ x
x(cot⁡2(x2)−1)cot⁡2(x2)+1+2cot⁡(x2)cot⁡2(x2)+12x\frac{\frac{\sqrt{x} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{2 \sqrt{x}} 
              ___   
  1         \/ x    
------ + -----------
csc(x)      /pi    \
         csc|-- - x|
            \2     /
--------------------
          ___       
      2*\/ x
xcsc⁡(−x+π2)+1csc⁡(x)2x\frac{\frac{\sqrt{x}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}}{2 \sqrt{x}} 
                ___ 
     1        \/ x  
----------- + ------
   /    pi\   sec(x)
sec|x - --|         
   \    2 /         
--------------------
          ___       
      2*\/ x
xsec⁡(x)+1sec⁡(x−π2)2x\frac{\frac{\sqrt{x}}{\sec{\left(x \right)}} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}{2 \sqrt{x}} 
       /x\      ___ /       2/x\\
  2*tan|-|    \/ x *|1 - tan |-||
       \2/          \        \2//
----------- + -------------------
       2/x\              2/x\    
1 + tan |-|       1 + tan |-|    
        \2/               \2/    
---------------------------------
                 ___             
             2*\/ x
x(1−tan⁡2(x2))tan⁡2(x2)+1+2tan⁡(x2)tan⁡2(x2)+12x\frac{\frac{\sqrt{x} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{2 \sqrt{x}} 
           ___ 
  1      \/ x  
------ + ------
csc(x)   sec(x)
---------------
        ___    
    2*\/ x
xsec⁡(x)+1csc⁡(x)2x\frac{\frac{\sqrt{x}}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}}{2 \sqrt{x}} 
  ___    /    pi\         
\/ x *sin|x + --| + sin(x)
         \    2 /         
--------------------------
             ___          
         2*\/ x
xsin⁡(x+π2)+sin⁡(x)2x\frac{\sqrt{x} \sin{\left(x + \frac{\pi}{2} \right)} + \sin{\left(x \right)}}{2 \sqrt{x}} 
  ___             /    pi\
\/ x *cos(x) + cos|x - --|
                  \    2 /
--------------------------
             ___          
         2*\/ x
xcos⁡(x)+cos⁡(x−π2)2x\frac{\sqrt{x} \cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{x}} 
   1    /  ___                \
-------*\\/ x *cos(x) + sin(x)/
    ___                        
2*\/ x
12x(xcos⁡(x)+sin⁡(x))\frac{1}{2 \sqrt{x}} \left(\sqrt{x} \cos{\left(x \right)} + \sin{\left(x \right)}\right) 
(1/(2*sqrt(x)))*(sqrt(x)*cos(x) + sin(x))
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