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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x-3/2)/(x^2+1)-2*x*sin(x-3/2)/(x^2+1)^2

¿Cómo vas a descomponer esta cos(x-3/2)/(x^2+1)-2*x*sin(x-3/2)/(x^2+1)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(x - 3/2)   2*x*sin(x - 3/2)
------------ - ----------------
    2                     2    
   x  + 1         / 2    \     
                  \x  + 1/
−2xsin⁡(x−32)(x2+1)2+cos⁡(x−32)x2+1- \frac{2 x \sin{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{3}{2} \right)}}{x^{2} + 1} 
cos(x - 3/2)/(x^2 + 1) - (2*x)*sin(x - 3/2)/(x^2 + 1)^2
Simplificación general [src] 
/     2\                                  
\1 + x /*cos(-3/2 + x) - 2*x*sin(-3/2 + x)
------------------------------------------
                        2                 
                /     2\                  
                \1 + x /
−2xsin⁡(x−32)+(x2+1)cos⁡(x−32)(x2+1)2\frac{- 2 x \sin{\left(x - \frac{3}{2} \right)} + \left(x^{2} + 1\right) \cos{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} 
((1 + x^2)*cos(-3/2 + x) - 2*x*sin(-3/2 + x))/(1 + x^2)^2
Respuesta numérica [src] 
cos(x - 3/2)/(1.0 + x^2) - 2.0*x*sin(x - 3/2)/(1.0 + x^2)^2
cos(x - 3/2)/(1.0 + x^2) - 2.0*x*sin(x - 3/2)/(1.0 + x^2)^2
Denominador común [src] 
 2                                                  
x *cos(-3/2 + x) - 2*x*sin(-3/2 + x) + cos(-3/2 + x)
----------------------------------------------------
                        4      2                    
                   1 + x  + 2*x
x2cos⁡(x−32)−2xsin⁡(x−32)+cos⁡(x−32)x4+2x2+1\frac{x^{2} \cos{\left(x - \frac{3}{2} \right)} - 2 x \sin{\left(x - \frac{3}{2} \right)} + \cos{\left(x - \frac{3}{2} \right)}}{x^{4} + 2 x^{2} + 1} 
(x^2*cos(-3/2 + x) - 2*x*sin(-3/2 + x) + cos(-3/2 + x))/(1 + x^4 + 2*x^2)
Potencias [src] 
 I*(-3/2 + x)    I*(3/2 - x)                                       
e               e                                                  
------------- + ------------       /   I*(3/2 - x)    I*(-3/2 + x)\
      2              2         I*x*\- e            + e            /
---------------------------- + ------------------------------------
                2                                   2              
           1 + x                            /     2\               
                                            \1 + x /
ix(−ei(32−x)+ei(x−32))(x2+1)2+ei(32−x)2+ei(x−32)2x2+1\frac{i x \left(- e^{i \left(\frac{3}{2} - x\right)} + e^{i \left(x - \frac{3}{2}\right)}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{\frac{e^{i \left(\frac{3}{2} - x\right)}}{2} + \frac{e^{i \left(x - \frac{3}{2}\right)}}{2}}{x^{2} + 1} 
(exp(i*(-3/2 + x))/2 + exp(i*(3/2 - x))/2)/(1 + x^2) + i*x*(-exp(i*(3/2 - x)) + exp(i*(-3/2 + x)))/(1 + x^2)^2
Unión de expresiones racionales [src] 
/     2\    /-3 + 2*x\          /-3 + 2*x\
\1 + x /*cos|--------| - 2*x*sin|--------|
            \   2    /          \   2    /
------------------------------------------
                        2                 
                /     2\                  
                \1 + x /
−2xsin⁡(2x−32)+(x2+1)cos⁡(2x−32)(x2+1)2\frac{- 2 x \sin{\left(\frac{2 x - 3}{2} \right)} + \left(x^{2} + 1\right) \cos{\left(\frac{2 x - 3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} 
((1 + x^2)*cos((-3 + 2*x)/2) - 2*x*sin((-3 + 2*x)/2))/(1 + x^2)^2
Combinatoria [src] 
 2                                                  
x *cos(-3/2 + x) - 2*x*sin(-3/2 + x) + cos(-3/2 + x)
----------------------------------------------------
                             2                      
                     /     2\                       
                     \1 + x /
x2cos⁡(x−32)−2xsin⁡(x−32)+cos⁡(x−32)(x2+1)2\frac{x^{2} \cos{\left(x - \frac{3}{2} \right)} - 2 x \sin{\left(x - \frac{3}{2} \right)} + \cos{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} 
(x^2*cos(-3/2 + x) - 2*x*sin(-3/2 + x) + cos(-3/2 + x))/(1 + x^2)^2
Abrimos la expresión [src] 
cos(3/2)*cos(x)   sin(3/2)*sin(x)   2*x*cos(3/2)*sin(x)   2*x*cos(x)*sin(3/2)
--------------- + --------------- - ------------------- + -------------------
      2                 2                   4      2              4      2   
     x  + 1            x  + 1          1 + x  + 2*x          1 + x  + 2*x
−2xsin⁡(x)cos⁡(32)x4+2x2+1+2xsin⁡(32)cos⁡(x)x4+2x2+1+sin⁡(32)sin⁡(x)x2+1+cos⁡(32)cos⁡(x)x2+1- \frac{2 x \sin{\left(x \right)} \cos{\left(\frac{3}{2} \right)}}{x^{4} + 2 x^{2} + 1} + \frac{2 x \sin{\left(\frac{3}{2} \right)} \cos{\left(x \right)}}{x^{4} + 2 x^{2} + 1} + \frac{\sin{\left(\frac{3}{2} \right)} \sin{\left(x \right)}}{x^{2} + 1} + \frac{\cos{\left(\frac{3}{2} \right)} \cos{\left(x \right)}}{x^{2} + 1} 
cos(3/2)*cos(x)/(x^2 + 1) + sin(3/2)*sin(x)/(x^2 + 1) - 2*x*cos(3/2)*sin(x)/(1 + x^4 + 2*x^2) + 2*x*cos(x)*sin(3/2)/(1 + x^4 + 2*x^2)
Denominador racional [src] 
        2                                                       
/     2\                                         3              
\1 + x / *cos(-3/2 + x) - 2*x*sin(-3/2 + x) - 2*x *sin(-3/2 + x)
----------------------------------------------------------------
                                   3                            
                           /     2\                             
                           \1 + x /
−2x3sin⁡(x−32)−2xsin⁡(x−32)+(x2+1)2cos⁡(x−32)(x2+1)3\frac{- 2 x^{3} \sin{\left(x - \frac{3}{2} \right)} - 2 x \sin{\left(x - \frac{3}{2} \right)} + \left(x^{2} + 1\right)^{2} \cos{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{3}} 
((1 + x^2)^2*cos(-3/2 + x) - 2*x*sin(-3/2 + x) - 2*x^3*sin(-3/2 + x))/(1 + x^2)^3
Parte trigonométrica [src] 
            2/  3   x\                /  3   x\    /  3   x\
     1 - tan |- - + -|         4*x*sec|- - + -|*sin|- - + -|
             \  4   2/                \  4   2/    \  4   2/
---------------------------- - -----------------------------
/     2\ /       2/  3   x\\           2                    
\1 + x /*|1 + tan |- - + -||   /     2\  /       2/  3   x\\
         \        \  4   2//   \1 + x / *|1 + tan |- - + -||
                                         \        \  4   2//
−4xsin⁡(x2−34)sec⁡(x2−34)(x2+1)2(tan⁡2(x2−34)+1)+1−tan⁡2(x2−34)(x2+1)(tan⁡2(x2−34)+1)- \frac{4 x \sin{\left(\frac{x}{2} - \frac{3}{4} \right)} \sec{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} 
          1                        2*x          
---------------------- - -----------------------
/     2\                         2              
\1 + x /*sec(-3/2 + x)   /     2\               
                         \1 + x / *csc(-3/2 + x)
−2x(x2+1)2csc⁡(x−32)+1(x2+1)sec⁡(x−32)- \frac{2 x}{\left(x^{2} + 1\right)^{2} \csc{\left(x - \frac{3}{2} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(x - \frac{3}{2} \right)}} 
                       /  3       pi\
                2*x*cos|- - + x - --|
cos(-3/2 + x)          \  2       2 /
------------- - ---------------------
         2                    2      
    1 + x             /     2\       
                      \1 + x /
−2xcos⁡(x−π2−32)(x2+1)2+cos⁡(x−32)x2+1- \frac{2 x \cos{\left(x - \frac{\pi}{2} - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{3}{2} \right)}}{x^{2} + 1} 
            2/  3   x\                       /  3   x\      
     1 - tan |- - + -|                4*x*tan|- - + -|      
             \  4   2/                       \  4   2/      
---------------------------- - -----------------------------
/     2\ /       2/  3   x\\           2                    
\1 + x /*|1 + tan |- - + -||   /     2\  /       2/  3   x\\
         \        \  4   2//   \1 + x / *|1 + tan |- - + -||
                                         \        \  4   2//
−4xtan⁡(x2−34)(x2+1)2(tan⁡2(x2−34)+1)+1−tan⁡2(x2−34)(x2+1)(tan⁡2(x2−34)+1)- \frac{4 x \tan{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} 
             2/  3   x\                      /  3   x\      
     -1 + cot |- - + -|               4*x*cot|- - + -|      
              \  4   2/                      \  4   2/      
---------------------------- - -----------------------------
/     2\ /       2/  3   x\\           2                    
\1 + x /*|1 + cot |- - + -||   /     2\  /       2/  3   x\\
         \        \  4   2//   \1 + x / *|1 + cot |- - + -||
                                         \        \  4   2//
−4xcot⁡(x2−34)(x2+1)2(cot⁡2(x2−34)+1)+cot⁡2(x2−34)−1(x2+1)(cot⁡2(x2−34)+1)- \frac{4 x \cot{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} - 1}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} 
                    /                 /  3   x\      /  3   x\\
                2*x*|cos(-3/2 + x)*tan|- - + -| + tan|- - + -||
cos(-3/2 + x)       \                 \  4   2/      \  4   2//
------------- - -----------------------------------------------
         2                                 2                   
    1 + x                          /     2\                    
                                   \1 + x /
−2x(cos⁡(x−32)tan⁡(x2−34)+tan⁡(x2−34))(x2+1)2+cos⁡(x−32)x2+1- \frac{2 x \left(\cos{\left(x - \frac{3}{2} \right)} \tan{\left(\frac{x}{2} - \frac{3}{4} \right)} + \tan{\left(\frac{x}{2} - \frac{3}{4} \right)}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{3}{2} \right)}}{x^{2} + 1} 
               2/  3   x\                                                               
            sec |- - + -|                                                               
                \  4   2/                                                               
      1 - ------------------                                                            
             2/  3   x   pi\                                 /  3   x\                  
          sec |- - + - - --|                          4*x*sec|- - + -|                  
              \  4   2   2 /                                 \  4   2/                  
--------------------------------- - ----------------------------------------------------
         /         2/  3   x\   \             /         2/  3   x\   \                  
         |      sec |- - + -|   |           2 |      sec |- - + -|   |                  
/     2\ |          \  4   2/   |   /     2\  |          \  4   2/   |    /  3   x   pi\
\1 + x /*|1 + ------------------|   \1 + x / *|1 + ------------------|*sec|- - + - - --|
         |       2/  3   x   pi\|             |       2/  3   x   pi\|    \  4   2   2 /
         |    sec |- - + - - --||             |    sec |- - + - - --||                  
         \        \  4   2   2 //             \        \  4   2   2 //
−4xsec⁡(x2−34)(x2+1)2(sec⁡2(x2−34)sec⁡2(x2−π2−34)+1)sec⁡(x2−π2−34)+−sec⁡2(x2−34)sec⁡2(x2−π2−34)+1(x2+1)(sec⁡2(x2−34)sec⁡2(x2−π2−34)+1)- \frac{4 x \sec{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + \frac{- \frac{\sec^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + 1}{\left(x^{2} + 1\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + 1\right)} 
               1                                                         
     1 - -------------                                                   
            2/  3   x\                                                   
         cot |- - + -|                                                   
             \  4   2/                            4*x                    
---------------------------- - ------------------------------------------
/     2\ /          1      \           2                                 
\1 + x /*|1 + -------------|   /     2\  /          1      \    /  3   x\
         |       2/  3   x\|   \1 + x / *|1 + -------------|*cot|- - + -|
         |    cot |- - + -||             |       2/  3   x\|    \  4   2/
         \        \  4   2//             |    cot |- - + -||             
                                         \        \  4   2//
−4x(1+1cot⁡2(x2−34))(x2+1)2cot⁡(x2−34)+1−1cot⁡2(x2−34)(1+1cot⁡2(x2−34))(x2+1)- \frac{4 x}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)^{2} \cot{\left(\frac{x}{2} - \frac{3}{4} \right)}} + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)} 
             2/3   pi   x\                                                     
          csc |- + -- - -|                                                     
              \4   2    2/                                                     
      1 - ----------------                                                     
              2/  3   x\                              /3   pi   x\             
           csc |- - + -|                       4*x*csc|- + -- - -|             
               \  4   2/                              \4   2    2/             
------------------------------- - ---------------------------------------------
         /       2/3   pi   x\\             /       2/3   pi   x\\             
         |    csc |- + -- - -||           2 |    csc |- + -- - -||             
/     2\ |        \4   2    2/|   /     2\  |        \4   2    2/|    /  3   x\
\1 + x /*|1 + ----------------|   \1 + x / *|1 + ----------------|*csc|- - + -|
         |        2/  3   x\  |             |        2/  3   x\  |    \  4   2/
         |     csc |- - + -|  |             |     csc |- - + -|  |             
         \         \  4   2/  /             \         \  4   2/  /
−4xcsc⁡(−x2+34+π2)(1+csc⁡2(−x2+34+π2)csc⁡2(x2−34))(x2+1)2csc⁡(x2−34)+1−csc⁡2(−x2+34+π2)csc⁡2(x2−34)(1+csc⁡2(−x2+34+π2)csc⁡2(x2−34))(x2+1)- \frac{4 x \csc{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)^{2} \csc{\left(\frac{x}{2} - \frac{3}{4} \right)}} + \frac{1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)} 
   /  3       pi\                    
sin|- - + x + --|                    
   \  2       2 /   2*x*sin(-3/2 + x)
----------------- - -----------------
           2                    2    
      1 + x             /     2\     
                        \1 + x /
−2xsin⁡(x−32)(x2+1)2+sin⁡(x−32+π2)x2+1- \frac{2 x \sin{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(x - \frac{3}{2} + \frac{\pi}{2} \right)}}{x^{2} + 1} 
              4/  3   x\                                                      
         4*sin |- - + -|                                                      
               \  4   2/                                                      
     1 - ---------------                              2/  3   x\              
             2                                 8*x*sin |- - + -|              
          sin (-3/2 + x)                               \  4   2/              
------------------------------ - ---------------------------------------------
         /         4/  3   x\\             /         4/  3   x\\              
         |    4*sin |- - + -||           2 |    4*sin |- - + -||              
/     2\ |          \  4   2/|   /     2\  |          \  4   2/|              
\1 + x /*|1 + ---------------|   \1 + x / *|1 + ---------------|*sin(-3/2 + x)
         |        2          |             |        2          |              
         \     sin (-3/2 + x)/             \     sin (-3/2 + x)/
−8xsin⁡2(x2−34)(x2+1)2(4sin⁡4(x2−34)sin⁡2(x−32)+1)sin⁡(x−32)+−4sin⁡4(x2−34)sin⁡2(x−32)+1(x2+1)(4sin⁡4(x2−34)sin⁡2(x−32)+1)- \frac{8 x \sin^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\frac{4 \sin^{4}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sin^{2}{\left(x - \frac{3}{2} \right)}} + 1\right) \sin{\left(x - \frac{3}{2} \right)}} + \frac{- \frac{4 \sin^{4}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sin^{2}{\left(x - \frac{3}{2} \right)}} + 1}{\left(x^{2} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sin^{2}{\left(x - \frac{3}{2} \right)}} + 1\right)} 
             2/  3   x   pi\                                                       
          cos |- - + - - --|                                                       
              \  4   2   2 /                                                       
      1 - ------------------                                                       
               2/  3   x\                               /  3   x   pi\             
            cos |- - + -|                        4*x*cos|- - + - - --|             
                \  4   2/                               \  4   2   2 /             
--------------------------------- - -----------------------------------------------
         /       2/  3   x   pi\\             /       2/  3   x   pi\\             
         |    cos |- - + - - --||           2 |    cos |- - + - - --||             
/     2\ |        \  4   2   2 /|   /     2\  |        \  4   2   2 /|    /  3   x\
\1 + x /*|1 + ------------------|   \1 + x / *|1 + ------------------|*cos|- - + -|
         |         2/  3   x\   |             |         2/  3   x\   |    \  4   2/
         |      cos |- - + -|   |             |      cos |- - + -|   |             
         \          \  4   2/   /             \          \  4   2/   /
−4xcos⁡(x2−π2−34)(1+cos⁡2(x2−π2−34)cos⁡2(x2−34))(x2+1)2cos⁡(x2−34)+1−cos⁡2(x2−π2−34)cos⁡2(x2−34)(1+cos⁡2(x2−π2−34)cos⁡2(x2−34))(x2+1)- \frac{4 x \cos{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)^{2} \cos{\left(\frac{x}{2} - \frac{3}{4} \right)}} + \frac{1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)} 
           1                         2*x          
------------------------ - -----------------------
/     2\    /3   pi    \           2              
\1 + x /*csc|- + -- - x|   /     2\               
            \2   2     /   \1 + x / *csc(-3/2 + x)
−2x(x2+1)2csc⁡(x−32)+1(x2+1)csc⁡(−x+32+π2)- \frac{2 x}{\left(x^{2} + 1\right)^{2} \csc{\left(x - \frac{3}{2} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(- x + \frac{3}{2} + \frac{\pi}{2} \right)}} 
          1                          2*x            
---------------------- - ---------------------------
/     2\                         2                  
\1 + x /*sec(-3/2 + x)   /     2\     /  3       pi\
                         \1 + x / *sec|- - + x - --|
                                      \  2       2 /
−2x(x2+1)2sec⁡(x−π2−32)+1(x2+1)sec⁡(x−32)- \frac{2 x}{\left(x^{2} + 1\right)^{2} \sec{\left(x - \frac{\pi}{2} - \frac{3}{2} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(x - \frac{3}{2} \right)}} 
1/((1 + x^2)*sec(-3/2 + x)) - 2*x/((1 + x^2)^2*sec(-3/2 + x - pi/2))
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