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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x)/(x-sin(x))+(-1+cos(x))*(2-cos(x))/(x-sin(x))^2

¿Cómo vas a descomponer esta sin(x)/(x-sin(x))+(-1+cos(x))*(2-cos(x))/(x-sin(x))^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(x)     (-1 + cos(x))*(2 - cos(x))
---------- + --------------------------
x - sin(x)                     2       
                   (x - sin(x))
(2−cos⁡(x))(cos⁡(x)−1)(x−sin⁡(x))2+sin⁡(x)x−sin⁡(x)\frac{\left(2 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right)}{\left(x - \sin{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{x - \sin{\left(x \right)}} 
sin(x)/(x - sin(x)) + ((-1 + cos(x))*(2 - cos(x)))/(x - sin(x))^2
Simplificación general [src] 
-3 + 3*cos(x) + x*sin(x)
------------------------
                 2      
     (x - sin(x))
xsin⁡(x)+3cos⁡(x)−3(x−sin⁡(x))2\frac{x \sin{\left(x \right)} + 3 \cos{\left(x \right)} - 3}{\left(x - \sin{\left(x \right)}\right)^{2}} 
(-3 + 3*cos(x) + x*sin(x))/(x - sin(x))^2
Respuesta numérica [src] 
sin(x)/(x - sin(x)) + (2.0 - cos(x))*(-1.0 + cos(x))/(x - sin(x))^2
sin(x)/(x - sin(x)) + (2.0 - cos(x))*(-1.0 + cos(x))/(x - sin(x))^2
Denominador racional [src] 
            2                                                 
(x - sin(x)) *sin(x) + (-1 + cos(x))*(2 - cos(x))*(x - sin(x))
--------------------------------------------------------------
                                    3                         
                        (x - sin(x))
(2−cos⁡(x))(x−sin⁡(x))(cos⁡(x)−1)+(x−sin⁡(x))2sin⁡(x)(x−sin⁡(x))3\frac{\left(2 - \cos{\left(x \right)}\right) \left(x - \sin{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right) + \left(x - \sin{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{\left(x - \sin{\left(x \right)}\right)^{3}} 
((x - sin(x))^2*sin(x) + (-1 + cos(x))*(2 - cos(x))*(x - sin(x)))/(x - sin(x))^3
Potencias [src] 
/      I*x    -I*x\ /     I*x    -I*x\                             
|     e      e    | |    e      e    |                             
|-1 + ---- + -----|*|2 - ---- - -----|         /   -I*x    I*x\    
\      2       2  / \     2       2  /       I*\- e     + e   /    
-------------------------------------- - --------------------------
                              2            /      /   -I*x    I*x\\
      /      /   -I*x    I*x\\             |    I*\- e     + e   /|
      |    I*\- e     + e   /|           2*|x + ------------------|
      |x + ------------------|             \            2         /
      \            2         /
−i(eix−e−ix)2(x+i(eix−e−ix)2)+(−eix2+2−e−ix2)(eix2−1+e−ix2)(x+i(eix−e−ix)2)2- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(x + \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)} + \frac{\left(- \frac{e^{i x}}{2} + 2 - \frac{e^{- i x}}{2}\right) \left(\frac{e^{i x}}{2} - 1 + \frac{e^{- i x}}{2}\right)}{\left(x + \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)^{2}} 
(-1 + exp(i*x)/2 + exp(-i*x)/2)*(2 - exp(i*x)/2 - exp(-i*x)/2)/(x + i*(-exp(-i*x) + exp(i*x))/2)^2 - i*(-exp(-i*x) + exp(i*x))/(2*(x + i*(-exp(-i*x) + exp(i*x))/2))
Combinatoria [src] 
        2         2                         
-2 - cos (x) - sin (x) + 3*cos(x) + x*sin(x)
--------------------------------------------
                           2                
               (x - sin(x))
xsin⁡(x)−sin⁡2(x)−cos⁡2(x)+3cos⁡(x)−2(x−sin⁡(x))2\frac{x \sin{\left(x \right)} - \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 2}{\left(x - \sin{\left(x \right)}\right)^{2}} 
(-2 - cos(x)^2 - sin(x)^2 + 3*cos(x) + x*sin(x))/(x - sin(x))^2
Unión de expresiones racionales [src] 
(-1 + cos(x))*(2 - cos(x)) + (x - sin(x))*sin(x)
------------------------------------------------
                             2                  
                 (x - sin(x))
(2−cos⁡(x))(cos⁡(x)−1)+(x−sin⁡(x))sin⁡(x)(x−sin⁡(x))2\frac{\left(2 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right) + \left(x - \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\left(x - \sin{\left(x \right)}\right)^{2}} 
((-1 + cos(x))*(2 - cos(x)) + (x - sin(x))*sin(x))/(x - sin(x))^2
Abrimos la expresión [src] 
                                                       2                                        
              2                 sin(x)              cos (x)                     3*cos(x)        
- ------------------------- + ---------- - ------------------------- + -------------------------
   2      2                   x - sin(x)    2      2                    2      2                
  x  + sin (x) - 2*x*sin(x)                x  + sin (x) - 2*x*sin(x)   x  + sin (x) - 2*x*sin(x)
−cos⁡2(x)x2−2xsin⁡(x)+sin⁡2(x)+3cos⁡(x)x2−2xsin⁡(x)+sin⁡2(x)−2x2−2xsin⁡(x)+sin⁡2(x)+sin⁡(x)x−sin⁡(x)- \frac{\cos^{2}{\left(x \right)}}{x^{2} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{x^{2} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} - \frac{2}{x^{2} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{x - \sin{\left(x \right)}} 
-2/(x^2 + sin(x)^2 - 2*x*sin(x)) + sin(x)/(x - sin(x)) - cos(x)^2/(x^2 + sin(x)^2 - 2*x*sin(x)) + 3*cos(x)/(x^2 + sin(x)^2 - 2*x*sin(x))
Parte trigonométrica [src] 
          /    pi\           
-3 + 3*sin|x + --| + x*sin(x)
          \    2 /           
-----------------------------
                    2        
        (x - sin(x))
xsin⁡(x)+3sin⁡(x+π2)−3(x−sin⁡(x))2\frac{x \sin{\left(x \right)} + 3 \sin{\left(x + \frac{\pi}{2} \right)} - 3}{\left(x - \sin{\left(x \right)}\right)^{2}} 
       3           x     
-3 + ------ + -----------
     sec(x)      /    pi\
              sec|x - --|
                 \    2 /
-------------------------
                     2   
    /         1     \    
    |x - -----------|    
    |       /    pi\|    
    |    sec|x - --||    
    \       \    2 //
xsec⁡(x−π2)−3+3sec⁡(x)(x−1sec⁡(x−π2))2\frac{\frac{x}{\sec{\left(x - \frac{\pi}{2} \right)}} - 3 + \frac{3}{\sec{\left(x \right)}}}{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} 
     /    pi\                               
  cos|x - --|                               
     \    2 /     (-1 + cos(x))*(2 - cos(x))
--------------- + --------------------------
       /    pi\                        2    
x - cos|x - --|       /       /    pi\\     
       \    2 /       |x - cos|x - --||     
                      \       \    2 //
(2−cos⁡(x))(cos⁡(x)−1)(x−cos⁡(x−π2))2+cos⁡(x−π2)x−cos⁡(x−π2)\frac{\left(2 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right)}{\left(x - \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x - \cos{\left(x - \frac{\pi}{2} \right)}} 
       /        2/x\\           /x\
     3*|-1 + cot |-||    2*x*cot|-|
       \         \2//           \2/
-3 + ---------------- + -----------
              2/x\             2/x\
       1 + cot |-|      1 + cot |-|
               \2/              \2/
-----------------------------------
                          2        
         /           /x\ \         
         |      2*cot|-| |         
         |           \2/ |         
         |x - -----------|         
         |           2/x\|         
         |    1 + cot |-||         
         \            \2//
2xcot⁡(x2)cot⁡2(x2)+1+3(cot⁡2(x2)−1)cot⁡2(x2)+1−3(x−2cot⁡(x2)cot⁡2(x2)+1)2\frac{\frac{2 x \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 3}{\left(x - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
          3          x   
-3 + ----------- + ------
        /pi    \   csc(x)
     csc|-- - x|         
        \2     /         
-------------------------
                  2      
      /      1   \       
      |x - ------|       
      \    csc(x)/
xcsc⁡(x)−3+3csc⁡(−x+π2)(x−1csc⁡(x))2\frac{\frac{x}{\csc{\left(x \right)}} - 3 + \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(x - \frac{1}{\csc{\left(x \right)}}\right)^{2}} 
/             2/x\\ /            2/x\\                                  
|     -1 + cot |-|| |    -1 + cot |-||                                  
|              \2/| |             \2/|                                  
|-1 + ------------|*|2 - ------------|                                  
|            2/x\ | |           2/x\ |                    /x\           
|     1 + cot |-| | |    1 + cot |-| |               2*cot|-|           
\             \2/ / \            \2/ /                    \2/           
-------------------------------------- + -------------------------------
                           2                           /           /x\ \
          /           /x\ \                            |      2*cot|-| |
          |      2*cot|-| |              /       2/x\\ |           \2/ |
          |           \2/ |              |1 + cot |-||*|x - -----------|
          |x - -----------|              \        \2// |           2/x\|
          |           2/x\|                            |    1 + cot |-||
          |    1 + cot |-||                            \            \2//
          \            \2//
2cot⁡(x2)(x−2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)+(−cot⁡2(x2)−1cot⁡2(x2)+1+2)(cot⁡2(x2)−1cot⁡2(x2)+1−1)(x−2cot⁡(x2)cot⁡2(x2)+1)2\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(x - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 2\right) \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)}{\left(x - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
                      /       1   \ /      1   \
                      |-1 + ------|*|2 - ------|
         1            \     sec(x)/ \    sec(x)/
------------------- + --------------------------
/      1   \                            2       
|x - ------|*csc(x)         /      1   \        
\    csc(x)/                |x - ------|        
                            \    csc(x)/
(−1+1sec⁡(x))(2−1sec⁡(x))(x−1csc⁡(x))2+1(x−1csc⁡(x))csc⁡(x)\frac{\left(-1 + \frac{1}{\sec{\left(x \right)}}\right) \left(2 - \frac{1}{\sec{\left(x \right)}}\right)}{\left(x - \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{1}{\left(x - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} 
             /        /    pi\\ /       /    pi\\
             |-1 + sin|x + --||*|2 - sin|x + --||
  sin(x)     \        \    2 // \       \    2 //
---------- + ------------------------------------
x - sin(x)                          2            
                        (x - sin(x))
(2−sin⁡(x+π2))(sin⁡(x+π2)−1)(x−sin⁡(x))2+sin⁡(x)x−sin⁡(x)\frac{\left(2 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} - 1\right)}{\left(x - \sin{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{x - \sin{\left(x \right)}} 
/            2/x\\ /           2/x\\                                  
|     1 - tan |-|| |    1 - tan |-||                                  
|             \2/| |            \2/|                                  
|-1 + -----------|*|2 - -----------|                                  
|            2/x\| |           2/x\|                    /x\           
|     1 + tan |-|| |    1 + tan |-||               2*tan|-|           
\             \2// \            \2//                    \2/           
------------------------------------ + -------------------------------
                          2                          /           /x\ \
         /           /x\ \                           |      2*tan|-| |
         |      2*tan|-| |             /       2/x\\ |           \2/ |
         |           \2/ |             |1 + tan |-||*|x - -----------|
         |x - -----------|             \        \2// |           2/x\|
         |           2/x\|                           |    1 + tan |-||
         |    1 + tan |-||                           \            \2//
         \            \2//
2tan⁡(x2)(x−2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)+(−1−tan⁡2(x2)tan⁡2(x2)+1+2)(1−tan⁡2(x2)tan⁡2(x2)+1−1)(x−2tan⁡(x2)tan⁡2(x2)+1)2\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(x - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 2\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)}{\left(x - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
                     /    pi\
-3 + 3*cos(x) + x*cos|x - --|
                     \    2 /
-----------------------------
                       2     
      /       /    pi\\      
      |x - cos|x - --||      
      \       \    2 //
xcos⁡(x−π2)+3cos⁡(x)−3(x−cos⁡(x−π2))2\frac{x \cos{\left(x - \frac{\pi}{2} \right)} + 3 \cos{\left(x \right)} - 3}{\left(x - \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}} 
-3 + 3*cos(x) + x*sin(x)
------------------------
                 2      
     (x - sin(x))
xsin⁡(x)+3cos⁡(x)−3(x−sin⁡(x))2\frac{x \sin{\left(x \right)} + 3 \cos{\left(x \right)} - 3}{\left(x - \sin{\left(x \right)}\right)^{2}} 
       /       2/x\\           /x\
     3*|1 - tan |-||    2*x*tan|-|
       \        \2//           \2/
-3 + --------------- + -----------
              2/x\            2/x\
       1 + tan |-|     1 + tan |-|
               \2/             \2/
----------------------------------
                         2        
        /           /x\ \         
        |      2*tan|-| |         
        |           \2/ |         
        |x - -----------|         
        |           2/x\|         
        |    1 + tan |-||         
        \            \2//
2xtan⁡(x2)tan⁡2(x2)+1+3(1−tan⁡2(x2))tan⁡2(x2)+1−3(x−2tan⁡(x2)tan⁡2(x2)+1)2\frac{\frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 3}{\left(x - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
                                /       1   \ /      1   \
                                |-1 + ------|*|2 - ------|
              1                 \     sec(x)/ \    sec(x)/
----------------------------- + --------------------------
/         1     \    /    pi\                        2    
|x - -----------|*sec|x - --|       /         1     \     
|       /    pi\|    \    2 /       |x - -----------|     
|    sec|x - --||                   |       /    pi\|     
\       \    2 //                   |    sec|x - --||     
                                    \       \    2 //
(−1+1sec⁡(x))(2−1sec⁡(x))(x−1sec⁡(x−π2))2+1(x−1sec⁡(x−π2))sec⁡(x−π2)\frac{\left(-1 + \frac{1}{\sec{\left(x \right)}}\right) \left(2 - \frac{1}{\sec{\left(x \right)}}\right)}{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} + \frac{1}{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
                      /          1     \ /         1     \
                      |-1 + -----------|*|2 - -----------|
                      |        /pi    \| |       /pi    \|
                      |     csc|-- - x|| |    csc|-- - x||
         1            \        \2     // \       \2     //
------------------- + ------------------------------------
/      1   \                                 2            
|x - ------|*csc(x)              /      1   \             
\    csc(x)/                     |x - ------|             
                                 \    csc(x)/
(−1+1csc⁡(−x+π2))(2−1csc⁡(−x+π2))(x−1csc⁡(x))2+1(x−1csc⁡(x))csc⁡(x)\frac{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(2 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)}{\left(x - \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{1}{\left(x - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} 
1/((x - 1/csc(x))*csc(x)) + (-1 + 1/csc(pi/2 - x))*(2 - 1/csc(pi/2 - x))/(x - 1/csc(x))^2
Denominador común [src] 
        2         2                         
-2 - cos (x) - sin (x) + 3*cos(x) + x*sin(x)
--------------------------------------------
          2      2                          
         x  + sin (x) - 2*x*sin(x)
xsin⁡(x)−sin⁡2(x)−cos⁡2(x)+3cos⁡(x)−2x2−2xsin⁡(x)+sin⁡2(x)\frac{x \sin{\left(x \right)} - \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 2}{x^{2} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} 
(-2 - cos(x)^2 - sin(x)^2 + 3*cos(x) + x*sin(x))/(x^2 + sin(x)^2 - 2*x*sin(x))
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