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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)
xcos⁡(2x)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))
Respuesta numérica [src] 
x*cos(2*x)/(-cos(x) + sin(x))
x*cos(2*x)/(-cos(x) + sin(x))
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
x(e2ix2+e−2ix2)−i(eix−e−ix)2−eix2−e−ix2\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)
2xcos⁡2(x)sin⁡(x)−cos⁡(x)−xsin⁡(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))
Combinatoria [src] 
  -x*cos(2*x)   
----------------
-sin(x) + cos(x)
−xcos⁡(2x)−sin⁡(x)+cos⁡(x)- \frac{x \cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}} 
-x*cos(2*x)/(-sin(x) + cos(x))
Parte trigonométrica [src] 
      ___  
 -x*\/ 2   
-----------
   /    pi\
sec|x - --|
   \    4 /
−2xsec⁡(x−π4)- \frac{\sqrt{2} x}{\sec{\left(x - \frac{\pi}{4} \right)}} 
                       ___                           
                   x*\/ 2 *cos(2*x)                  
-----------------------------------------------------
                    ___________    ___________       
    ___            /       ___    /       ___        
- \/ 2 *cos(x) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(x)
2xcos⁡(2x)2−22+2sin⁡(x)−2cos⁡(x)\frac{\sqrt{2} x \cos{\left(2 x \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(x \right)} - \sqrt{2} \cos{\left(x \right)}} 
               x               
-------------------------------
/     1          1   \         
|----------- - ------|*sec(2*x)
|   /    pi\   sec(x)|         
|sec|x - --|         |         
\   \    2 /         /
x(1sec⁡(x−π2)−1sec⁡(x))sec⁡(2x)\frac{x}{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}} 
      x*cos(2*x)     
---------------------
             /    pi\
-cos(x) + cos|x - --|
             \    2 /
xcos⁡(2x)−cos⁡(x)+cos⁡(x−π2)\frac{x \cos{\left(2 x \right)}}{- \cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}} 
     ___    2    /       2/x   pi\\ /       2   \ 
-x*\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/ 
                 \        \2   8 //               
--------------------------------------------------
                 /       2/x   pi\\               
               2*|1 - tan |- + --||               
                 \        \2   8 //
−2x(1−tan⁡2(x))(tan⁡2(x2+π8)+1)cos⁡2(x)2(1−tan⁡2(x2+π8))- \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)}}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)} 
     ___             /     pi\ 
-x*\/ 2 *cos(2*x)*csc|-x + --| 
                     \     4 / 
-------------------------------
               2
−2xcos⁡(2x)csc⁡(−x+π4)2- \frac{\sqrt{2} x \cos{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{4} \right)}}{2} 
      ___  
 -x*\/ 2   
-----------
   /    pi\
csc|x + --|
   \    4 /
−2xcsc⁡(x+π4)- \frac{\sqrt{2} x}{\csc{\left(x + \frac{\pi}{4} \right)}} 
     ___    /pi      \ 
-x*\/ 2 *sin|-- + 2*x| 
            \2       / 
-----------------------
         /    3*pi\    
    2*sin|x + ----|    
         \     4  /
−2xsin⁡(2x+π2)2sin⁡(x+3π4)- \frac{\sqrt{2} x \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{3 \pi}{4} \right)}} 
     ___ /       2/x   pi\\ /        2   \ 
-x*\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/ 
         \        \2   8 //                
-------------------------------------------
      /       2   \ /        2/x   pi\\    
    2*\1 + cot (x)/*|-1 + cot |- + --||    
                    \         \2   8 //
−2x(cot⁡2(x)−1)(cot⁡2(x2+π8)+1)2(cot⁡2(x)+1)(cot⁡2(x2+π8)−1)- \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)}{2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)} 
       ___    /x   pi\
-2*x*\/ 2 *tan|- + --|
              \2   8 /
----------------------
          2/x   pi\   
   1 + tan |- + --|   
           \2   8 /
−22xtan⁡(x2+π8)tan⁡2(x2+π8)+1- \frac{2 \sqrt{2} x \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} 
            x             
--------------------------
/  1        1   \         
|------ - ------|*sec(2*x)
\csc(x)   sec(x)/
x(−1sec⁡(x)+1csc⁡(x))sec⁡(2x)\frac{x}{\left(- \frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}} 
       ___    2/x   pi\    /x   pi\
-2*x*\/ 2 *sin |- + --|*cot|- + --|
               \2   8 /    \2   8 /
−22xsin⁡2(x2+π8)cot⁡(x2+π8)- 2 \sqrt{2} x \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)} 
     ___    /     pi\ 
-x*\/ 2 *csc|-x + --| 
            \     4 / 
----------------------
        /pi      \    
   2*csc|-- - 2*x|    
        \2       /
−2xcsc⁡(−x+π4)2csc⁡(−2x+π2)- \frac{\sqrt{2} x \csc{\left(- x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
     ___          
-x*\/ 2 *cos(2*x) 
------------------
       /    pi\   
  2*cos|x + --|   
       \    4 /
−2xcos⁡(2x)2cos⁡(x+π4)- \frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}} 
        /pi      \    
   x*sin|-- + 2*x|    
        \2       /    
----------------------
     /    pi\         
- sin|x + --| + sin(x)
     \    2 /
xsin⁡(2x+π2)sin⁡(x)−sin⁡(x+π2)\frac{x \sin{\left(2 x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} - \sin{\left(x + \frac{\pi}{2} \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//
x(cot⁡2(x)−1)(−cot⁡2(x2)−1cot⁡2(x2)+1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x)+1)\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)} 
     ___    /    pi\
-x*\/ 2 *sin|x + --|
            \    4 /
−2xsin⁡(x+π4)- \sqrt{2} x \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//
x(1−tan⁡2(x))(−1−tan⁡2(x2)tan⁡2(x2)+1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x)+1)\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   pi\
-2*x*\/ 2 *cot|- + --|
              \2   8 /
----------------------
          2/x   pi\   
   1 + cot |- + --|   
           \2   8 /
−22xcot⁡(x2+π8)cot⁡2(x2+π8)+1- \frac{2 \sqrt{2} x \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} 
                 x                  
------------------------------------
/  1           1     \    /pi      \
|------ - -----------|*csc|-- - 2*x|
|csc(x)      /pi    \|    \2       /
|         csc|-- - x||              
\            \2     //
x(−1csc⁡(−x+π2)+1csc⁡(x))csc⁡(−2x+π2)\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)}} 
     ___ /       2/x   pi\\ /       2   \ 
-x*\/ 2 *|1 + tan |- + --||*\1 - tan (x)/ 
         \        \2   8 //               
------------------------------------------
      /       2   \ /       2/x   pi\\    
    2*\1 + tan (x)/*|1 - tan |- + --||    
                    \        \2   8 //
−2x(1−tan⁡2(x))(tan⁡2(x2+π8)+1)2(1−tan⁡2(x2+π8))(tan⁡2(x)+1)- \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)}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)} 
     ___             /    pi\ 
-x*\/ 2 *cos(2*x)*sec|x + --| 
                     \    4 / 
------------------------------
              2
−2xcos⁡(2x)sec⁡(x+π4)2- \frac{\sqrt{2} x \cos{\left(2 x \right)} \sec{\left(x + \frac{\pi}{4} \right)}}{2} 
     ___    /    pi\
-x*\/ 2 *cos|x - --|
            \    4 /
−2xcos⁡(x−π4)- \sqrt{2} x \cos{\left(x - \frac{\pi}{4} \right)} 
     ___    /    pi\ 
-x*\/ 2 *sec|x + --| 
            \    4 / 
---------------------
      2*sec(2*x)
−2xsec⁡(x+π4)2sec⁡(2x)- \frac{\sqrt{2} x \sec{\left(x + \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}} 
-x*sqrt(2)*sec(x + pi/4)/(2*sec(2*x))
Denominador común [src] 
  -x*cos(2*x)   
----------------
-sin(x) + cos(x)
−xcos⁡(2x)−sin⁡(x)+cos⁡(x)- \frac{x \cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}} 
-x*cos(2*x)/(-sin(x) + cos(x))
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