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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
 cos(x)    sin(x)
-------- - ------
       2   1 - x 
(1 - x)
$$\frac{\cos{\left(x \right)}}{\left(1 - x\right)^{2}} - \frac{\sin{\left(x \right)}}{1 - x}$$ 
cos(x)/(1 - x)^2 - sin(x)/(1 - x)
Simplificación general [src] 
(-1 + x)*sin(x) + cos(x)
------------------------
               2        
       (-1 + x)
$$\frac{\left(x - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(x - 1\right)^{2}}$$ 
((-1 + x)*sin(x) + cos(x))/(-1 + x)^2
Denominador racional [src] 
                        2       
(1 - x)*cos(x) - (1 - x) *sin(x)
--------------------------------
                   3            
            (1 - x)
$$\frac{- \left(1 - x\right)^{2} \sin{\left(x \right)} + \left(1 - x\right) \cos{\left(x \right)}}{\left(1 - x\right)^{3}}$$ 
((1 - x)*cos(x) - (1 - x)^2*sin(x))/(1 - x)^3
Potencias [src] 
 I*x    -I*x                     
e      e                         
---- + -----     /   -I*x    I*x\
 2       2     I*\- e     + e   /
------------ + ------------------
         2         2*(1 - x)     
  (1 - x)
$$\frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(1 - x\right)} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\left(1 - x\right)^{2}}$$ 
 cos(x)    sin(x)
-------- - ------
       2   1 - x 
(1 - x)
$$- \frac{\sin{\left(x \right)}}{1 - x} + \frac{\cos{\left(x \right)}}{\left(1 - x\right)^{2}}$$ 
cos(x)/(1 - x)^2 - sin(x)/(1 - x)
Denominador común [src] 
-sin(x) + x*sin(x) + cos(x)
---------------------------
             2             
        1 + x  - 2*x
$$\frac{x \sin{\left(x \right)} - \sin{\left(x \right)} + \cos{\left(x \right)}}{x^{2} - 2 x + 1}$$ 
(-sin(x) + x*sin(x) + cos(x))/(1 + x^2 - 2*x)
Unión de expresiones racionales [src] 
-(1 - x)*sin(x) + cos(x)
------------------------
               2        
        (1 - x)
$$\frac{- \left(1 - x\right) \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(1 - x\right)^{2}}$$ 
(-(1 - x)*sin(x) + cos(x))/(1 - x)^2
Abrimos la expresión [src] 
   cos(x)      sin(x)
------------ - ------
     2         1 - x 
1 + x  - 2*x
$$\frac{\cos{\left(x \right)}}{x^{2} - 2 x + 1} - \frac{\sin{\left(x \right)}}{1 - x}$$ 
cos(x)/(1 + x^2 - 2*x) - sin(x)/(1 - x)
Parte trigonométrica [src] 
            2/x\                     /x\      
     1 - tan |-|                2*tan|-|      
             \2/                     \2/      
---------------------- - ---------------------
/       2/x\\        2   /       2/x\\        
|1 + tan |-||*(1 - x)    |1 + tan |-||*(1 - x)
\        \2//            \        \2//
$$- \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 - x\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 - x\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
   /    pi\         
sin|x + --|         
   \    2 /   sin(x)
----------- - ------
         2    1 - x 
  (1 - x)
$$- \frac{\sin{\left(x \right)}}{1 - x} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\left(1 - x\right)^{2}}$$ 
 cos(x)    sin(x)
-------- - ------
       2   1 - x 
(1 - x)
$$- \frac{\sin{\left(x \right)}}{1 - x} + \frac{\cos{\left(x \right)}}{\left(1 - x\right)^{2}}$$ 
              /    pi\
           cos|x - --|
 cos(x)       \    2 /
-------- - -----------
       2      1 - x   
(1 - x)
$$- \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{1 - x} + \frac{\cos{\left(x \right)}}{\left(1 - x\right)^{2}}$$ 
             2/x\                    /x\      
     -1 + cot |-|               2*cot|-|      
              \2/                    \2/      
---------------------- - ---------------------
/       2/x\\        2   /       2/x\\        
|1 + cot |-||*(1 - x)    |1 + cot |-||*(1 - x)
\        \2//            \        \2//
$$- \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 - x\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 - x\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
       1                1       
--------------- - --------------
       2          (1 - x)*csc(x)
(1 - x) *sec(x)
$$- \frac{1}{\left(1 - x\right) \csc{\left(x \right)}} + \frac{1}{\left(1 - x\right)^{2} \sec{\left(x \right)}}$$ 
         1                   1       
-------------------- - --------------
       2    /pi    \   (1 - x)*csc(x)
(1 - x) *csc|-- - x|                 
            \2     /
$$- \frac{1}{\left(1 - x\right) \csc{\left(x \right)}} + \frac{1}{\left(1 - x\right)^{2} \csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
       1                   1         
--------------- - -------------------
       2                     /    pi\
(1 - x) *sec(x)   (1 - x)*sec|x - --|
                             \    2 /
$$- \frac{1}{\left(1 - x\right) \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - x\right)^{2} \sec{\left(x \right)}}$$ 
1/((1 - x)^2*sec(x)) - 1/((1 - x)*sec(x - pi/2))
Combinatoria [src] 
-sin(x) + x*sin(x) + cos(x)
---------------------------
                 2         
         (-1 + x)
$$\frac{x \sin{\left(x \right)} - \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(x - 1\right)^{2}}$$ 
(-sin(x) + x*sin(x) + cos(x))/(-1 + x)^2
Respuesta numérica [src] 
cos(x)/(1.0 - x)^2 - sin(x)/(1.0 - x)
cos(x)/(1.0 - x)^2 - sin(x)/(1.0 - x)
Compilar la expresión [src] 
 cos(x)    sin(x)
-------- - ------
       2   1 - x 
(1 - x)
$$- \frac{\sin{\left(x \right)}}{1 - x} + \frac{\cos{\left(x \right)}}{\left(1 - x\right)^{2}}$$ 
cos(x)/(1 - x)^2 - sin(x)/(1 - x)
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