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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(x)     sin(x) 
------ - ---------
log(x)        2   
         x*log (x)
cos⁡(x)log⁡(x)−sin⁡(x)xlog⁡(x)2\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}} 
cos(x)/log(x) - sin(x)/(x*log(x)^2)
Denominador racional [src] 
                      2          
-log(x)*sin(x) + x*log (x)*cos(x)
---------------------------------
                 3               
            x*log (x)
xlog⁡(x)2cos⁡(x)−log⁡(x)sin⁡(x)xlog⁡(x)3\frac{x \log{\left(x \right)}^{2} \cos{\left(x \right)} - \log{\left(x \right)} \sin{\left(x \right)}}{x \log{\left(x \right)}^{3}} 
(-log(x)*sin(x) + x*log(x)^2*cos(x))/(x*log(x)^3)
Potencias [src] 
 I*x    -I*x                     
e      e                         
---- + -----     /   -I*x    I*x\
 2       2     I*\- e     + e   /
------------ + ------------------
   log(x)                2       
                  2*x*log (x)
eix2+e−ix2log⁡(x)+i(eix−e−ix)2xlog⁡(x)2\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\log{\left(x \right)}} + \frac{i \left(e^{i x} - e^{- i x}\right)}{2 x \log{\left(x \right)}^{2}} 
(exp(i*x)/2 + exp(-i*x)/2)/log(x) + i*(-exp(-i*x) + exp(i*x))/(2*x*log(x)^2)
Combinatoria [src] 
-sin(x) + x*cos(x)*log(x)
-------------------------
             2           
        x*log (x)
xlog⁡(x)cos⁡(x)−sin⁡(x)xlog⁡(x)2\frac{x \log{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}} 
(-sin(x) + x*cos(x)*log(x))/(x*log(x)^2)
Unión de expresiones racionales [src] 
-sin(x) + x*cos(x)*log(x)
-------------------------
             2           
        x*log (x)
xlog⁡(x)cos⁡(x)−sin⁡(x)xlog⁡(x)2\frac{x \log{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}} 
(-sin(x) + x*cos(x)*log(x))/(x*log(x)^2)
Respuesta numérica [src] 
cos(x)/log(x) - sin(x)/(x*log(x)^2)
cos(x)/log(x) - sin(x)/(x*log(x)^2)
Parte trigonométrica [src] 
        2/x\               /x\       
-1 + cot |-|          2*cot|-|       
         \2/               \2/       
------------ - ----------------------
       2/x\      /       2/x\\       
1 + cot |-|    x*|1 + cot |-||*log(x)
        \2/      \        \2//       
-------------------------------------
                log(x)
cot⁡2(x2)−1cot⁡2(x2)+1−2cot⁡(x2)x(cot⁡2(x2)+1)log⁡(x)log⁡(x)\frac{\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)}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}}}{\log{\left(x \right)}} 
       2/x\               /x\       
1 - tan |-|          2*tan|-|       
        \2/               \2/       
----------- - ----------------------
       2/x\     /       2/x\\       
1 + tan |-|   x*|1 + tan |-||*log(x)
        \2/     \        \2//       
------------------------------------
               log(x)
1−tan⁡2(x2)tan⁡2(x2)+1−2tan⁡(x2)x(tan⁡2(x2)+1)log⁡(x)log⁡(x)\frac{\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)}}{x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}}}{\log{\left(x \right)}} 
            /    pi\
         cos|x - --|
cos(x)      \    2 /
------ - -----------
log(x)         2    
          x*log (x)
cos⁡(x)log⁡(x)−cos⁡(x−π2)xlog⁡(x)2\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x \log{\left(x \right)}^{2}} 
            2/x\                    /x\       
    -1 + cot |-|               2*cot|-|       
             \2/                    \2/       
-------------------- - -----------------------
/       2/x\\            /       2/x\\    2   
|1 + cot |-||*log(x)   x*|1 + cot |-||*log (x)
\        \2//            \        \2//
cot⁡2(x2)−1(cot⁡2(x2)+1)log⁡(x)−2cot⁡(x2)x(cot⁡2(x2)+1)log⁡(x)2\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}^{2}} 
   sin(x)       /    pi\
- -------- + sin|x + --|
  x*log(x)      \    2 /
------------------------
         log(x)
sin⁡(x+π2)−sin⁡(x)xlog⁡(x)log⁡(x)\frac{\sin{\left(x + \frac{\pi}{2} \right)} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}}}{\log{\left(x \right)}} 
      1                   1          
------------- - ---------------------
log(x)*sec(x)        2       /    pi\
                x*log (x)*sec|x - --|
                             \    2 /
1log⁡(x)sec⁡(x)−1xlog⁡(x)2sec⁡(x−π2)\frac{1}{\log{\left(x \right)} \sec{\left(x \right)}} - \frac{1}{x \log{\left(x \right)}^{2} \sec{\left(x - \frac{\pi}{2} \right)}} 
   /    pi\            
sin|x + --|            
   \    2 /     sin(x) 
----------- - ---------
   log(x)          2   
              x*log (x)
sin⁡(x+π2)log⁡(x)−sin⁡(x)xlog⁡(x)2\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}} 
     /    pi\         
  cos|x - --|         
     \    2 /         
- ----------- + cos(x)
    x*log(x)          
----------------------
        log(x)
cos⁡(x)−cos⁡(x−π2)xlog⁡(x)log⁡(x)\frac{\cos{\left(x \right)} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x \log{\left(x \right)}}}{\log{\left(x \right)}} 
      1                1        
------------- - ----------------
log(x)*sec(x)               2   
                x*csc(x)*log (x)
1log⁡(x)sec⁡(x)−1xlog⁡(x)2csc⁡(x)\frac{1}{\log{\left(x \right)} \sec{\left(x \right)}} - \frac{1}{x \log{\left(x \right)}^{2} \csc{\left(x \right)}} 
     1               1       
----------- - ---------------
   /pi    \   x*csc(x)*log(x)
csc|-- - x|                  
   \2     /                  
-----------------------------
            log(x)
1csc⁡(−x+π2)−1xlog⁡(x)csc⁡(x)log⁡(x)\frac{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{x \log{\left(x \right)} \csc{\left(x \right)}}}{\log{\left(x \right)}} 
        1                   1        
------------------ - ----------------
   /pi    \                      2   
csc|-- - x|*log(x)   x*csc(x)*log (x)
   \2     /
1log⁡(x)csc⁡(−x+π2)−1xlog⁡(x)2csc⁡(x)\frac{1}{\log{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{x \log{\left(x \right)}^{2} \csc{\left(x \right)}} 
  1               1          
------ - --------------------
sec(x)               /    pi\
         x*log(x)*sec|x - --|
                     \    2 /
-----------------------------
            log(x)
1sec⁡(x)−1xlog⁡(x)sec⁡(x−π2)log⁡(x)\frac{\frac{1}{\sec{\left(x \right)}} - \frac{1}{x \log{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}}{\log{\left(x \right)}} 
           2/x\                     /x\       
    1 - tan |-|                2*tan|-|       
            \2/                     \2/       
-------------------- - -----------------------
/       2/x\\            /       2/x\\    2   
|1 + tan |-||*log(x)   x*|1 + tan |-||*log (x)
\        \2//            \        \2//
1−tan⁡2(x2)(tan⁡2(x2)+1)log⁡(x)−2tan⁡(x2)x(tan⁡2(x2)+1)log⁡(x)2\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}^{2}} 
   sin(x)          
- -------- + cos(x)
  x*log(x)         
-------------------
       log(x)
cos⁡(x)−sin⁡(x)xlog⁡(x)log⁡(x)\frac{\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}}}{\log{\left(x \right)}} 
(-sin(x)/(x*log(x)) + cos(x))/log(x)
Denominador común [src] 
-sin(x) + x*cos(x)*log(x)
-------------------------
             2           
        x*log (x)
xlog⁡(x)cos⁡(x)−sin⁡(x)xlog⁡(x)2\frac{x \log{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}} 
(-sin(x) + x*cos(x)*log(x))/(x*log(x)^2)
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