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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)-2xsin(x-3/2)/(x^2+1)^2 expresión en fracciones?

Solución

Ha introducido [src]

cos(x - 3/2)   2*x*sin(x - 3/2)
------------ - ----------------
    2                     2    
   x  + 1         / 2    \     
                  \x  + 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 /

$$\frac{- 2 x \sin{\left(x - \frac{3}{2} \right)} + \left(x^{2} + 1\right) \cos{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}}$$

((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

$$\frac{x^{2} \cos{\left(x - \frac{3}{2} \right)} - 2 x \sin{\left(x - \frac{3}{2} \right)} + \cos{\left(x - \frac{3}{2} \right)}}{x^{4} + 2 x^{2} + 1}$$

(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 /

$$\frac{i x \left(- e^{i \left(\frac{3}{2} - x\right)} + e^{i \left(x - \frac{3}{2}\right)}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{\frac{e^{i \left(\frac{3}{2} - x\right)}}{2} + \frac{e^{i \left(x - \frac{3}{2}\right)}}{2}}{x^{2} + 1}$$

(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 /

$$\frac{- 2 x \sin{\left(\frac{2 x - 3}{2} \right)} + \left(x^{2} + 1\right) \cos{\left(\frac{2 x - 3}{2} \right)}}{\left(x^{2} + 1\right)^{2}}$$

((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 /

$$\frac{x^{2} \cos{\left(x - \frac{3}{2} \right)} - 2 x \sin{\left(x - \frac{3}{2} \right)} + \cos{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}}$$

(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

$$- \frac{2 x \sin{\left(x \right)} \cos{\left(\frac{3}{2} \right)}}{x^{4} + 2 x^{2} + 1} + \frac{2 x \sin{\left(\frac{3}{2} \right)} \cos{\left(x \right)}}{x^{4} + 2 x^{2} + 1} + \frac{\sin{\left(\frac{3}{2} \right)} \sin{\left(x \right)}}{x^{2} + 1} + \frac{\cos{\left(\frac{3}{2} \right)} \cos{\left(x \right)}}{x^{2} + 1}$$

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 /

$$\frac{- 2 x^{3} \sin{\left(x - \frac{3}{2} \right)} - 2 x \sin{\left(x - \frac{3}{2} \right)} + \left(x^{2} + 1\right)^{2} \cos{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{3}}$$

((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//

$$- \frac{4 x \sin{\left(\frac{x}{2} - \frac{3}{4} \right)} \sec{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)}$$

          1                        2*x          
---------------------- - -----------------------
/     2\                         2              
\1 + x /*sec(-3/2 + x)   /     2\               
                         \1 + x / *csc(-3/2 + x)

$$- \frac{2 x}{\left(x^{2} + 1\right)^{2} \csc{\left(x - \frac{3}{2} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(x - \frac{3}{2} \right)}}$$

                       /  3       pi\
                2*x*cos|- - + x - --|
cos(-3/2 + x)          \  2       2 /
------------- - ---------------------
         2                    2      
    1 + x             /     2\       
                      \1 + x /

$$- \frac{2 x \cos{\left(x - \frac{\pi}{2} - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{3}{2} \right)}}{x^{2} + 1}$$

            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//

$$- \frac{4 x \tan{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)}$$

             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//

$$- \frac{4 x \cot{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} - 1}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)} + 1\right)}$$

                    /                 /  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 /

$$- \frac{2 x \left(\cos{\left(x - \frac{3}{2} \right)} \tan{\left(\frac{x}{2} - \frac{3}{4} \right)} + \tan{\left(\frac{x}{2} - \frac{3}{4} \right)}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{3}{2} \right)}}{x^{2} + 1}$$

               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 //

$$- \frac{4 x \sec{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + \frac{- \frac{\sec^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + 1}{\left(x^{2} + 1\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}} + 1\right)}$$

               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//

$$- \frac{4 x}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)^{2} \cot{\left(\frac{x}{2} - \frac{3}{4} \right)}} + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)}$$

             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/  /

$$- \frac{4 x \csc{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)^{2} \csc{\left(\frac{x}{2} - \frac{3}{4} \right)}} + \frac{1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{3}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)}$$

   /  3       pi\                    
sin|- - + x + --|                    
   \  2       2 /   2*x*sin(-3/2 + x)
----------------- - -----------------
           2                    2    
      1 + x             /     2\     
                        \1 + x /

$$- \frac{2 x \sin{\left(x - \frac{3}{2} \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(x - \frac{3}{2} + \frac{\pi}{2} \right)}}{x^{2} + 1}$$

              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)/

$$- \frac{8 x \sin^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\left(x^{2} + 1\right)^{2} \left(\frac{4 \sin^{4}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sin^{2}{\left(x - \frac{3}{2} \right)}} + 1\right) \sin{\left(x - \frac{3}{2} \right)}} + \frac{- \frac{4 \sin^{4}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sin^{2}{\left(x - \frac{3}{2} \right)}} + 1}{\left(x^{2} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} - \frac{3}{4} \right)}}{\sin^{2}{\left(x - \frac{3}{2} \right)}} + 1\right)}$$

             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/   /

$$- \frac{4 x \cos{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)^{2} \cos{\left(\frac{x}{2} - \frac{3}{4} \right)}} + \frac{1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} - \frac{3}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{3}{4} \right)}}\right) \left(x^{2} + 1\right)}$$

           1                         2*x          
------------------------ - -----------------------
/     2\    /3   pi    \           2              
\1 + x /*csc|- + -- - x|   /     2\               
            \2   2     /   \1 + x / *csc(-3/2 + x)

$$- \frac{2 x}{\left(x^{2} + 1\right)^{2} \csc{\left(x - \frac{3}{2} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(- x + \frac{3}{2} + \frac{\pi}{2} \right)}}$$

          1                          2*x            
---------------------- - ---------------------------
/     2\                         2                  
\1 + x /*sec(-3/2 + x)   /     2\     /  3       pi\
                         \1 + x / *sec|- - + x - --|
                                      \  2       2 /

$$- \frac{2 x}{\left(x^{2} + 1\right)^{2} \sec{\left(x - \frac{\pi}{2} - \frac{3}{2} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(x - \frac{3}{2} \right)}}$$

1/((1 + x^2)*sec(-3/2 + x)) - 2*x/((1 + x^2)^2*sec(-3/2 + x - pi/2))

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¿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?

Solución

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