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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
            / 2    \                    
pi*sin(n)   \n  - 2/*sin(n) + 2*n*cos(n)
--------- - ----------------------------
    n                      3            
                       pi*n
−2ncos⁡(n)+(n2−2)sin⁡(n)πn3+πsin⁡(n)n- \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} + \frac{\pi \sin{\left(n \right)}}{n} 
(pi*sin(n))/n - ((n^2 - 2)*sin(n) + (2*n)*cos(n))/(pi*n^3)
Simplificación general [src] 
/     2\                         2  2       
\2 - n /*sin(n) - 2*n*cos(n) + pi *n *sin(n)
--------------------------------------------
                       3                    
                   pi*n
π2n2sin⁡(n)−2ncos⁡(n)+(2−n2)sin⁡(n)πn3\frac{\pi^{2} n^{2} \sin{\left(n \right)} - 2 n \cos{\left(n \right)} + \left(2 - n^{2}\right) \sin{\left(n \right)}}{\pi n^{3}} 
((2 - n^2)*sin(n) - 2*n*cos(n) + pi^2*n^2*sin(n))/(pi*n^3)
Respuesta numérica [src] 
3.14159265358979*sin(n)/n - 0.318309886183791*((-2.0 + n^2)*sin(n) + 2.0*n*cos(n))/n^3
3.14159265358979*sin(n)/n - 0.318309886183791*((-2.0 + n^2)*sin(n) + 2.0*n*cos(n))/n^3
Denominador racional [src] 
  /  /      2\                    \     2  3       
n*\- \-2 + n /*sin(n) - 2*n*cos(n)/ + pi *n *sin(n)
---------------------------------------------------
                           4                       
                       pi*n
π2n3sin⁡(n)+n(−2ncos⁡(n)−(n2−2)sin⁡(n))πn4\frac{\pi^{2} n^{3} \sin{\left(n \right)} + n \left(- 2 n \cos{\left(n \right)} - \left(n^{2} - 2\right) \sin{\left(n \right)}\right)}{\pi n^{4}} 
(n*(-(-2 + n^2)*sin(n) - 2*n*cos(n)) + pi^2*n^3*sin(n))/(pi*n^4)
Unión de expresiones racionales [src] 
  /      2\                         2  2       
- \-2 + n /*sin(n) - 2*n*cos(n) + pi *n *sin(n)
-----------------------------------------------
                         3                     
                     pi*n
π2n2sin⁡(n)−2ncos⁡(n)−(n2−2)sin⁡(n)πn3\frac{\pi^{2} n^{2} \sin{\left(n \right)} - 2 n \cos{\left(n \right)} - \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
(-(-2 + n^2)*sin(n) - 2*n*cos(n) + pi^2*n^2*sin(n))/(pi*n^3)
Potencias [src] 
              /      2\                    
pi*sin(n)   - \-2 + n /*sin(n) - 2*n*cos(n)
--------- + -------------------------------
    n                        3             
                         pi*n
πsin⁡(n)n+−2ncos⁡(n)−(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} + \frac{- 2 n \cos{\left(n \right)} - \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
            /      2\                    
pi*sin(n)   \-2 + n /*sin(n) + 2*n*cos(n)
--------- - -----------------------------
    n                       3            
                        pi*n
πsin⁡(n)n−2ncos⁡(n)+(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
      / I*n    -I*n\     /      2\ /   -I*n    I*n\                        
      |e      e    |   I*\-2 + n /*\- e     + e   /                        
  2*n*|---- + -----| - ----------------------------        /   -I*n    I*n\
      \ 2       2  /                2                 pi*I*\- e     + e   /
- ------------------------------------------------- - ---------------------
                            3                                  2*n         
                        pi*n
−iπ(ein−e−in)2n−2n(ein2+e−in2)−i(n2−2)(ein−e−in)2πn3- \frac{i \pi \left(e^{i n} - e^{- i n}\right)}{2 n} - \frac{2 n \left(\frac{e^{i n}}{2} + \frac{e^{- i n}}{2}\right) - \frac{i \left(n^{2} - 2\right) \left(e^{i n} - e^{- i n}\right)}{2}}{\pi n^{3}} 
-(2*n*(exp(i*n)/2 + exp(-i*n)/2) - i*(-2 + n^2)*(-exp(-i*n) + exp(i*n))/2)/(pi*n^3) - pi*i*(-exp(-i*n) + exp(i*n))/(2*n)
Compilar la expresión [src] 
            /      2\                    
pi*sin(n)   \-2 + n /*sin(n) + 2*n*cos(n)
--------- - -----------------------------
    n                       3            
                        pi*n
πsin⁡(n)n−2ncos⁡(n)+(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
pi*sin(n)/n - ((-2 + n^2)*sin(n) + 2*n*cos(n))/(pi*n^3)
Denominador común [src] 
            2                         2  2       
2*sin(n) - n *sin(n) - 2*n*cos(n) + pi *n *sin(n)
-------------------------------------------------
                          3                      
                      pi*n
−n2sin⁡(n)+π2n2sin⁡(n)−2ncos⁡(n)+2sin⁡(n)πn3\frac{- n^{2} \sin{\left(n \right)} + \pi^{2} n^{2} \sin{\left(n \right)} - 2 n \cos{\left(n \right)} + 2 \sin{\left(n \right)}}{\pi n^{3}} 
(2*sin(n) - n^2*sin(n) - 2*n*cos(n) + pi^2*n^2*sin(n))/(pi*n^3)
Combinatoria [src] 
            2                         2  2       
2*sin(n) - n *sin(n) - 2*n*cos(n) + pi *n *sin(n)
-------------------------------------------------
                          3                      
                      pi*n
−n2sin⁡(n)+π2n2sin⁡(n)−2ncos⁡(n)+2sin⁡(n)πn3\frac{- n^{2} \sin{\left(n \right)} + \pi^{2} n^{2} \sin{\left(n \right)} - 2 n \cos{\left(n \right)} + 2 \sin{\left(n \right)}}{\pi n^{3}} 
(2*sin(n) - n^2*sin(n) - 2*n*cos(n) + pi^2*n^2*sin(n))/(pi*n^3)
Abrimos la expresión [src] 
            / 2    \                    
pi*sin(n)   \n  - 2/*sin(n) + 2*n*cos(n)
--------- - ----------------------------
    n                      3            
                       pi*n
πsin⁡(n)n−2ncos⁡(n)+(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
pi*sin(n)   sin(n)   2*cos(n)   2*sin(n)
--------- - ------ - -------- + --------
    n        pi*n         2          3  
                      pi*n       pi*n
πsin⁡(n)n−sin⁡(n)πn−2cos⁡(n)πn2+2sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{\sin{\left(n \right)}}{\pi n} - \frac{2 \cos{\left(n \right)}}{\pi n^{2}} + \frac{2 \sin{\left(n \right)}}{\pi n^{3}} 
(pi*sin(n))/n - sin(n)/(pi*n) - 2*cos(n)/(pi*n^2) + 2*sin(n)/(pi*n^3)
Parte trigonométrica [src] 
            / 2    \                    
pi*sin(n)   \n  - 2/*sin(n) + 2*n*cos(n)
--------- - ----------------------------
    n                      3            
                       pi*n
πsin⁡(n)n−2ncos⁡(n)+(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
                   2              
           -4 + 2*n        2*n    
           --------- + -----------
            2*csc(n)      /pi    \
                       csc|-- - n|
   pi                     \2     /
-------- - -----------------------
n*csc(n)                3         
                    pi*n
πncsc⁡(n)−2ncsc⁡(−n+π2)+2n2−42csc⁡(n)πn3\frac{\pi}{n \csc{\left(n \right)}} - \frac{\frac{2 n}{\csc{\left(- n + \frac{\pi}{2} \right)}} + \frac{2 n^{2} - 4}{2 \csc{\left(n \right)}}}{\pi n^{3}} 
            /        2\                    
            \-4 + 2*n /*sin(n)             
            ------------------ + 2*n*cos(n)
pi*sin(n)           2                      
--------- - -------------------------------
    n                        3             
                         pi*n
πsin⁡(n)n−2ncos⁡(n)+(2n2−4)sin⁡(n)2πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \frac{\left(2 n^{2} - 4\right) \sin{\left(n \right)}}{2}}{\pi n^{3}} 
      /       2/n\\     /      2\    /n\                  
  2*n*|1 - tan |-||   2*\-2 + n /*tan|-|                  
      \        \2//                  \2/                  
  ----------------- + ------------------                  
            2/n\                2/n\                 /n\  
     1 + tan |-|         1 + tan |-|         2*pi*tan|-|  
             \2/                 \2/                 \2/  
- -------------------------------------- + ---------------
                      3                      /       2/n\\
                  pi*n                     n*|1 + tan |-||
                                             \        \2//
2πtan⁡(n2)n(tan⁡2(n2)+1)−2n(1−tan⁡2(n2))tan⁡2(n2)+1+2(n2−2)tan⁡(n2)tan⁡2(n2)+1πn3\frac{2 \pi \tan{\left(\frac{n}{2} \right)}}{n \left(\tan^{2}{\left(\frac{n}{2} \right)} + 1\right)} - \frac{\frac{2 n \left(1 - \tan^{2}{\left(\frac{n}{2} \right)}\right)}{\tan^{2}{\left(\frac{n}{2} \right)} + 1} + \frac{2 \left(n^{2} - 2\right) \tan{\left(\frac{n}{2} \right)}}{\tan^{2}{\left(\frac{n}{2} \right)} + 1}}{\pi n^{3}} 
                 2         
           -2 + n     2*n  
           ------- + ------
   pi       csc(n)   sec(n)
-------- - ----------------
n*csc(n)            3      
                pi*n
πncsc⁡(n)−2nsec⁡(n)+n2−2csc⁡(n)πn3\frac{\pi}{n \csc{\left(n \right)}} - \frac{\frac{2 n}{\sec{\left(n \right)}} + \frac{n^{2} - 2}{\csc{\left(n \right)}}}{\pi n^{3}} 
                 /        2\    /    pi\             
                 \-4 + 2*n /*cos|n - --|             
      /    pi\                  \    2 /             
pi*cos|n - --|   ----------------------- + 2*n*cos(n)
      \    2 /              2                        
-------------- - ------------------------------------
      n                             3                
                                pi*n
πcos⁡(n−π2)n−2ncos⁡(n)+(2n2−4)cos⁡(n−π2)2πn3\frac{\pi \cos{\left(n - \frac{\pi}{2} \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \frac{\left(2 n^{2} - 4\right) \cos{\left(n - \frac{\pi}{2} \right)}}{2}}{\pi n^{3}} 
      /        2/n\\     /      2\    /n\                  
  2*n*|-1 + cot |-||   2*\-2 + n /*cot|-|                  
      \         \2//                  \2/                  
  ------------------ + ------------------                  
            2/n\                 2/n\                 /n\  
     1 + cot |-|          1 + cot |-|         2*pi*cot|-|  
             \2/                  \2/                 \2/  
- --------------------------------------- + ---------------
                       3                      /       2/n\\
                   pi*n                     n*|1 + cot |-||
                                              \        \2//
2πcot⁡(n2)n(cot⁡2(n2)+1)−2n(cot⁡2(n2)−1)cot⁡2(n2)+1+2(n2−2)cot⁡(n2)cot⁡2(n2)+1πn3\frac{2 \pi \cot{\left(\frac{n}{2} \right)}}{n \left(\cot^{2}{\left(\frac{n}{2} \right)} + 1\right)} - \frac{\frac{2 n \left(\cot^{2}{\left(\frac{n}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{n}{2} \right)} + 1} + \frac{2 \left(n^{2} - 2\right) \cot{\left(\frac{n}{2} \right)}}{\cot^{2}{\left(\frac{n}{2} \right)} + 1}}{\pi n^{3}} 
                 2              
           -2 + n        2*n    
           ------- + -----------
            csc(n)      /pi    \
                     csc|-- - n|
   pi                   \2     /
-------- - ---------------------
n*csc(n)               3        
                   pi*n
πncsc⁡(n)−2ncsc⁡(−n+π2)+n2−2csc⁡(n)πn3\frac{\pi}{n \csc{\left(n \right)}} - \frac{\frac{2 n}{\csc{\left(- n + \frac{\pi}{2} \right)}} + \frac{n^{2} - 2}{\csc{\left(n \right)}}}{\pi n^{3}} 
            /      2\                 /    pi\
            \-2 + n /*sin(n) + 2*n*sin|n + --|
pi*sin(n)                             \    2 /
--------- - ----------------------------------
    n                         3               
                          pi*n
πsin⁡(n)n−2nsin⁡(n+π2)+(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \sin{\left(n + \frac{\pi}{2} \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
      /    pi\   /      2\    /    pi\             
pi*cos|n - --|   \-2 + n /*cos|n - --| + 2*n*cos(n)
      \    2 /                \    2 /             
-------------- - ----------------------------------
      n                            3               
                               pi*n
πcos⁡(n−π2)n−2ncos⁡(n)+(n2−2)cos⁡(n−π2)πn3\frac{\pi \cos{\left(n - \frac{\pi}{2} \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \cos{\left(n - \frac{\pi}{2} \right)}}{\pi n^{3}} 
  /        2\    /n\       /        2/n\\                  
  \-4 + 2*n /*cot|-|   2*n*|-1 + cot |-||                  
                 \2/       \         \2//                  
  ------------------ + ------------------                  
            2/n\                 2/n\                 /n\  
     1 + cot |-|          1 + cot |-|         2*pi*cot|-|  
             \2/                  \2/                 \2/  
- --------------------------------------- + ---------------
                       3                      /       2/n\\
                   pi*n                     n*|1 + cot |-||
                                              \        \2//
2πcot⁡(n2)n(cot⁡2(n2)+1)−2n(cot⁡2(n2)−1)cot⁡2(n2)+1+(2n2−4)cot⁡(n2)cot⁡2(n2)+1πn3\frac{2 \pi \cot{\left(\frac{n}{2} \right)}}{n \left(\cot^{2}{\left(\frac{n}{2} \right)} + 1\right)} - \frac{\frac{2 n \left(\cot^{2}{\left(\frac{n}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{n}{2} \right)} + 1} + \frac{\left(2 n^{2} - 4\right) \cot{\left(\frac{n}{2} \right)}}{\cot^{2}{\left(\frac{n}{2} \right)} + 1}}{\pi n^{3}} 
                          2           
                  -4 + 2*n       2*n  
                ------------- + ------
                     /    pi\   sec(n)
                2*sec|n - --|         
      pi             \    2 /         
------------- - ----------------------
     /    pi\               3         
n*sec|n - --|           pi*n          
     \    2 /
πnsec⁡(n−π2)−2nsec⁡(n)+2n2−42sec⁡(n−π2)πn3\frac{\pi}{n \sec{\left(n - \frac{\pi}{2} \right)}} - \frac{\frac{2 n}{\sec{\left(n \right)}} + \frac{2 n^{2} - 4}{2 \sec{\left(n - \frac{\pi}{2} \right)}}}{\pi n^{3}} 
            /      2\                    
pi*sin(n)   \-2 + n /*sin(n) + 2*n*cos(n)
--------- - -----------------------------
    n                       3            
                        pi*n
πsin⁡(n)n−2ncos⁡(n)+(n2−2)sin⁡(n)πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \cos{\left(n \right)} + \left(n^{2} - 2\right) \sin{\left(n \right)}}{\pi n^{3}} 
                        2           
                  -2 + n       2*n  
                ----------- + ------
                   /    pi\   sec(n)
                sec|n - --|         
      pi           \    2 /         
------------- - --------------------
     /    pi\              3        
n*sec|n - --|          pi*n         
     \    2 /
πnsec⁡(n−π2)−2nsec⁡(n)+n2−2sec⁡(n−π2)πn3\frac{\pi}{n \sec{\left(n - \frac{\pi}{2} \right)}} - \frac{\frac{2 n}{\sec{\left(n \right)}} + \frac{n^{2} - 2}{\sec{\left(n - \frac{\pi}{2} \right)}}}{\pi n^{3}} 
            /        2\                         
            \-4 + 2*n /*sin(n)          /    pi\
            ------------------ + 2*n*sin|n + --|
pi*sin(n)           2                   \    2 /
--------- - ------------------------------------
    n                          3                
                           pi*n
πsin⁡(n)n−2nsin⁡(n+π2)+(2n2−4)sin⁡(n)2πn3\frac{\pi \sin{\left(n \right)}}{n} - \frac{2 n \sin{\left(n + \frac{\pi}{2} \right)} + \frac{\left(2 n^{2} - 4\right) \sin{\left(n \right)}}{2}}{\pi n^{3}} 
  /        2\    /n\       /       2/n\\                  
  \-4 + 2*n /*tan|-|   2*n*|1 - tan |-||                  
                 \2/       \        \2//                  
  ------------------ + -----------------                  
            2/n\                 2/n\                /n\  
     1 + tan |-|          1 + tan |-|        2*pi*tan|-|  
             \2/                  \2/                \2/  
- -------------------------------------- + ---------------
                      3                      /       2/n\\
                  pi*n                     n*|1 + tan |-||
                                             \        \2//
2πtan⁡(n2)n(tan⁡2(n2)+1)−2n(1−tan⁡2(n2))tan⁡2(n2)+1+(2n2−4)tan⁡(n2)tan⁡2(n2)+1πn3\frac{2 \pi \tan{\left(\frac{n}{2} \right)}}{n \left(\tan^{2}{\left(\frac{n}{2} \right)} + 1\right)} - \frac{\frac{2 n \left(1 - \tan^{2}{\left(\frac{n}{2} \right)}\right)}{\tan^{2}{\left(\frac{n}{2} \right)} + 1} + \frac{\left(2 n^{2} - 4\right) \tan{\left(\frac{n}{2} \right)}}{\tan^{2}{\left(\frac{n}{2} \right)} + 1}}{\pi n^{3}} 
-((-4 + 2*n^2)*tan(n/2)/(1 + tan(n/2)^2) + 2*n*(1 - tan(n/2)^2)/(1 + tan(n/2)^2))/(pi*n^3) + 2*pi*tan(n/2)/(n*(1 + tan(n/2)^2))
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