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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   cos(2*x)*x  
---------------
sin(x) + cos(x)
$$\frac{x \cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}$$ 
(cos(2*x)*x)/(sin(x) + cos(x))
Simplificación general [src] 
    ___         
x*\/ 2 *cos(2*x)
----------------
      /    pi\  
 2*sin|x + --|  
      \    4 /
$$\frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}}$$ 
x*sqrt(2)*cos(2*x)/(2*sin(x + pi/4))
Potencias [src] 
         / -2*I*x    2*I*x\      
         |e         e     |      
       x*|------- + ------|      
         \   2        2   /      
---------------------------------
 I*x    -I*x     /   -I*x    I*x\
e      e       I*\- e     + e   /
---- + ----- - ------------------
 2       2             2
$$\frac{x \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}$$ 
x*(exp(-2*i*x)/2 + exp(2*i*x)/2)/(exp(i*x)/2 + exp(-i*x)/2 - i*(-exp(-i*x) + exp(i*x))/2)
Abrimos la expresión [src] 
                             2     
         x            2*x*cos (x)  
- --------------- + ---------------
  sin(x) + cos(x)   sin(x) + cos(x)
$$\frac{2 x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} - \frac{x}{\sin{\left(x \right)} + \cos{\left(x \right)}}$$ 
-x/(sin(x) + cos(x)) + 2*x*cos(x)^2/(sin(x) + cos(x))
Respuesta numérica [src] 
x*cos(2*x)/(cos(x) + sin(x))
x*cos(2*x)/(cos(x) + sin(x))
Parte trigonométrica [src] 
    ___    2    /        2   \    /x   pi\
x*\/ 2 *sin (x)*\-1 + cot (x)/*tan|- + --|
                                  \2   8 /
------------------------------------------
                   2/x   pi\              
              4*sin |- + --|              
                    \2   8 /
$$\frac{\sqrt{2} x \left(\cot^{2}{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{4 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$ 
    ___    /    pi\
x*\/ 2 *csc|x + --|
           \    4 /
-------------------
       /pi      \  
  2*csc|-- - 2*x|  
       \2       /
$$\frac{\sqrt{2} x \csc{\left(x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$ 
    ___    /    pi\
x*\/ 2 *cos|x + --|
           \    4 /
$$\sqrt{2} x \cos{\left(x + \frac{\pi}{4} \right)}$$ 
    ___ /       2/x   pi\\ /       2   \
x*\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
        \        \2   8 //              
----------------------------------------
        /       2   \    /x   pi\       
      4*\1 + tan (x)/*tan|- + --|       
                         \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$ 
               /        2   \             
             x*\-1 + cot (x)/             
------------------------------------------
              /        2/x\          /x\ \
              |-1 + cot |-|     2*cot|-| |
/       2   \ |         \2/          \2/ |
\1 + cot (x)/*|------------ + -----------|
              |       2/x\           2/x\|
              |1 + cot |-|    1 + cot |-||
              \        \2/            \2//
$$\frac{x \left(\cot^{2}{\left(x \right)} - 1\right)}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$ 
                 x                  
------------------------------------
/  1           1     \    /pi      \
|------ + -----------|*csc|-- - 2*x|
|csc(x)      /pi    \|    \2       /
|         csc|-- - x||              
\            \2     //
$$\frac{x}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$ 
    ___    /pi      \
x*\/ 2 *sin|-- + 2*x|
           \2       /
---------------------
         /    pi\    
    2*sin|x + --|    
         \    4 /
$$\frac{\sqrt{2} x \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}}$$ 
               /       2   \             
             x*\1 - tan (x)/             
-----------------------------------------
              /       2/x\          /x\ \
              |1 - tan |-|     2*tan|-| |
/       2   \ |        \2/          \2/ |
\1 + tan (x)/*|----------- + -----------|
              |       2/x\          2/x\|
              |1 + tan |-|   1 + tan |-||
              \        \2/           \2//
$$\frac{x \left(1 - \tan^{2}{\left(x \right)}\right)}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$ 
    ___         
x*\/ 2 *cos(2*x)
----------------
      /    pi\  
 2*sin|x + --|  
      \    4 /
$$\frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}}$$ 
    ___ /        2/x   pi\\
x*\/ 2 *|-1 + cot |- + --||
        \         \2   8 //
---------------------------
             2/x   pi\     
      1 + cot |- + --|     
              \2   8 /
$$\frac{\sqrt{2} x \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}$$ 
    ___    /    3*pi\
x*\/ 2 *sin|x + ----|
           \     4  /
$$\sqrt{2} x \sin{\left(x + \frac{3 \pi}{4} \right)}$$ 
     x*cos(2*x)     
--------------------
            /    pi\
cos(x) + cos|x - --|
            \    2 /
$$\frac{x \cos{\left(2 x \right)}}{\cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}}$$ 
    ___    2/x   pi\ /        2/x   pi\\
x*\/ 2 *sin |- + --|*|-1 + cot |- + --||
            \2   8 / \         \2   8 //
$$\sqrt{2} x \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}$$ 
               x               
-------------------------------
/  1           1     \         
|------ + -----------|*sec(2*x)
|sec(x)      /    pi\|         
|         sec|x - --||         
\            \    2 //
$$\frac{x}{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}}$$ 
    ___             /    pi\
x*\/ 2 *cos(2*x)*sec|x - --|
                    \    4 /
----------------------------
             2
$$\frac{\sqrt{2} x \cos{\left(2 x \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{2}$$ 
    ___ /       2/x   pi\\
x*\/ 2 *|1 - tan |- + --||
        \        \2   8 //
--------------------------
            2/x   pi\     
     1 + tan |- + --|     
             \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}$$ 
    ___         
x*\/ 2 *cos(2*x)
----------------
      /    pi\  
 2*cos|x - --|  
      \    4 /
$$\frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \cos{\left(x - \frac{\pi}{4} \right)}}$$ 
            x             
--------------------------
/  1        1   \         
|------ + ------|*sec(2*x)
\csc(x)   sec(x)/
$$\frac{x}{\left(\frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}}$$ 
    ___    2    /       2/x   pi\\ /       2   \
x*\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/
                \        \2   8 //              
------------------------------------------------
                      /x   pi\                  
                 4*tan|- + --|                  
                      \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(x \right)}}{4 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$ 
      ___   
  x*\/ 2    
------------
   /     pi\
csc|-x + --|
   \     4 /
$$\frac{\sqrt{2} x}{\csc{\left(- x + \frac{\pi}{4} \right)}}$$ 
    ___             /    pi\
x*\/ 2 *cos(2*x)*csc|x + --|
                    \    4 /
----------------------------
             2
$$\frac{\sqrt{2} x \cos{\left(2 x \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{2}$$ 
    ___    /    pi\
x*\/ 2 *sec|x - --|
           \    4 /
-------------------
     2*sec(2*x)
$$\frac{\sqrt{2} x \sec{\left(x - \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}}$$ 
      ___  
  x*\/ 2   
-----------
   /    pi\
sec|x + --|
   \    4 /
$$\frac{\sqrt{2} x}{\sec{\left(x + \frac{\pi}{4} \right)}}$$ 
       /pi      \   
  x*sin|-- + 2*x|   
       \2       /   
--------------------
            /    pi\
sin(x) + sin|x + --|
            \    2 /
$$\frac{x \sin{\left(2 x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}}$$ 
    ___ /       2/x   pi\\ /        2   \
x*\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
        \        \2   8 //               
-----------------------------------------
         /       2   \    /x   pi\       
       4*\1 + cot (x)/*cot|- + --|       
                          \2   8 /
$$\frac{\sqrt{2} x \left(\cot^{2}{\left(x \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$ 
x*sqrt(2)*(1 + cot(x/2 + pi/8)^2)*(-1 + cot(x)^2)/(4*(1 + cot(x)^2)*cot(x/2 + pi/8))
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