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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x)/(x^2+3)-2*x*sin(x)/(x^2+3)^2

¿Cómo vas a descomponer esta cos(x)/(x^2+3)-2*x*sin(x)/(x^2+3)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(x)   2*x*sin(x)
------ - ----------
 2               2 
x  + 3   / 2    \  
         \x  + 3/
−2xsin⁡(x)(x2+3)2+cos⁡(x)x2+3- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} + 3} 
cos(x)/(x^2 + 3) - (2*x)*sin(x)/(x^2 + 3)^2
Simplificación general [src] 
/     2\                    
\3 + x /*cos(x) - 2*x*sin(x)
----------------------------
                 2          
         /     2\           
         \3 + x /
−2xsin⁡(x)+(x2+3)cos⁡(x)(x2+3)2\frac{- 2 x \sin{\left(x \right)} + \left(x^{2} + 3\right) \cos{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} 
((3 + x^2)*cos(x) - 2*x*sin(x))/(3 + x^2)^2
Respuesta numérica [src] 
cos(x)/(3.0 + x^2) - 0.222222222222222*x*sin(x)/(1 + 0.333333333333333*x^2)^2
cos(x)/(3.0 + x^2) - 0.222222222222222*x*sin(x)/(1 + 0.333333333333333*x^2)^2
Potencias [src] 
 I*x    -I*x                       
e      e                           
---- + -----       /   -I*x    I*x\
 2       2     I*x*\- e     + e   /
------------ + --------------------
        2                   2      
   3 + x            /     2\       
                    \3 + x /
ix(eix−e−ix)(x2+3)2+eix2+e−ix2x2+3\frac{i x \left(e^{i x} - e^{- i x}\right)}{\left(x^{2} + 3\right)^{2}} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{x^{2} + 3} 
(exp(i*x)/2 + exp(-i*x)/2)/(3 + x^2) + i*x*(-exp(-i*x) + exp(i*x))/(3 + x^2)^2
Abrimos la expresión [src] 
cos(x)     2*x*sin(x) 
------ - -------------
 2            4      2
x  + 3   9 + x  + 6*x
−2xsin⁡(x)x4+6x2+9+cos⁡(x)x2+3- \frac{2 x \sin{\left(x \right)}}{x^{4} + 6 x^{2} + 9} + \frac{\cos{\left(x \right)}}{x^{2} + 3} 
cos(x)/(x^2 + 3) - 2*x*sin(x)/(9 + x^4 + 6*x^2)
Denominador común [src] 
            2                    
3*cos(x) + x *cos(x) - 2*x*sin(x)
---------------------------------
               4      2          
          9 + x  + 6*x
x2cos⁡(x)−2xsin⁡(x)+3cos⁡(x)x4+6x2+9\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + 3 \cos{\left(x \right)}}{x^{4} + 6 x^{2} + 9} 
(3*cos(x) + x^2*cos(x) - 2*x*sin(x))/(9 + x^4 + 6*x^2)
Unión de expresiones racionales [src] 
/     2\                    
\3 + x /*cos(x) - 2*x*sin(x)
----------------------------
                 2          
         /     2\           
         \3 + x /
−2xsin⁡(x)+(x2+3)cos⁡(x)(x2+3)2\frac{- 2 x \sin{\left(x \right)} + \left(x^{2} + 3\right) \cos{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} 
((3 + x^2)*cos(x) - 2*x*sin(x))/(3 + x^2)^2
Parte trigonométrica [src] 
                /    pi\
         2*x*cos|x - --|
cos(x)          \    2 /
------ - ---------------
     2              2   
3 + x       /     2\    
            \3 + x /
−2xcos⁡(x−π2)(x2+3)2+cos⁡(x)x2+3- \frac{2 x \cos{\left(x - \frac{\pi}{2} \right)}}{\left(x^{2} + 3\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} + 3} 
            2/x\                       /x\      
     1 - tan |-|                4*x*tan|-|      
             \2/                       \2/      
---------------------- - -----------------------
/       2/x\\ /     2\                         2
|1 + tan |-||*\3 + x /   /       2/x\\ /     2\ 
\        \2//            |1 + tan |-||*\3 + x / 
                         \        \2//
−4xtan⁡(x2)(x2+3)2(tan⁡2(x2)+1)+1−tan⁡2(x2)(x2+3)(tan⁡2(x2)+1)- \frac{4 x \tan{\left(\frac{x}{2} \right)}}{\left(x^{2} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(x^{2} + 3\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
       1                2*x       
--------------- - ----------------
/     2\                  2       
\3 + x /*sec(x)   /     2\        
                  \3 + x / *csc(x)
−2x(x2+3)2csc⁡(x)+1(x2+3)sec⁡(x)- \frac{2 x}{\left(x^{2} + 3\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(x^{2} + 3\right) \sec{\left(x \right)}} 
         1                   2*x       
-------------------- - ----------------
/     2\    /pi    \           2       
\3 + x /*csc|-- - x|   /     2\        
            \2     /   \3 + x / *csc(x)
−2x(x2+3)2csc⁡(x)+1(x2+3)csc⁡(−x+π2)- \frac{2 x}{\left(x^{2} + 3\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(x^{2} + 3\right) \csc{\left(- x + \frac{\pi}{2} \right)}} 
   /    pi\             
sin|x + --|             
   \    2 /   2*x*sin(x)
----------- - ----------
        2             2 
   3 + x      /     2\  
              \3 + x /
−2xsin⁡(x)(x2+3)2+sin⁡(x+π2)x2+3- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{x^{2} + 3} 
             2/x\                      /x\      
     -1 + cot |-|               4*x*cot|-|      
              \2/                      \2/      
---------------------- - -----------------------
/       2/x\\ /     2\                         2
|1 + cot |-||*\3 + x /   /       2/x\\ /     2\ 
\        \2//            |1 + cot |-||*\3 + x / 
                         \        \2//
−4xcot⁡(x2)(x2+3)2(cot⁡2(x2)+1)+cot⁡2(x2)−1(x2+3)(cot⁡2(x2)+1)- \frac{4 x \cot{\left(\frac{x}{2} \right)}}{\left(x^{2} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(x^{2} + 3\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
       1                   2*x         
--------------- - ---------------------
/     2\                  2            
\3 + x /*sec(x)   /     2\     /    pi\
                  \3 + x / *sec|x - --|
                               \    2 /
−2x(x2+3)2sec⁡(x−π2)+1(x2+3)sec⁡(x)- \frac{2 x}{\left(x^{2} + 3\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(x^{2} + 3\right) \sec{\left(x \right)}} 
1/((3 + x^2)*sec(x)) - 2*x/((3 + x^2)^2*sec(x - pi/2))
Combinatoria [src] 
            2                    
3*cos(x) + x *cos(x) - 2*x*sin(x)
---------------------------------
                    2            
            /     2\             
            \3 + x /
x2cos⁡(x)−2xsin⁡(x)+3cos⁡(x)(x2+3)2\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + 3 \cos{\left(x \right)}}{\left(x^{2} + 3\right)^{2}} 
(3*cos(x) + x^2*cos(x) - 2*x*sin(x))/(3 + x^2)^2
Denominador racional [src] 
        2                             
/     2\               /     2\       
\3 + x / *cos(x) - 2*x*\3 + x /*sin(x)
--------------------------------------
                      3               
              /     2\                
              \3 + x /
−2x(x2+3)sin⁡(x)+(x2+3)2cos⁡(x)(x2+3)3\frac{- 2 x \left(x^{2} + 3\right) \sin{\left(x \right)} + \left(x^{2} + 3\right)^{2} \cos{\left(x \right)}}{\left(x^{2} + 3\right)^{3}} 
((3 + x^2)^2*cos(x) - 2*x*(3 + x^2)*sin(x))/(3 + x^2)^3
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