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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
             /u*pi\                /u*pi\    
          cos|----|             sin|----| - u
             \ 2  /                \ 2  /    
cos(u*x)*----------- + sin(u*x)*-------------
            /     2\                /     2\ 
         pi*\1 - u /             pi*\1 - u /
−u+sin⁡(πu2)π(1−u2)sin⁡(ux)+cos⁡(πu2)π(1−u2)cos⁡(ux)\frac{- u + \sin{\left(\frac{\pi u}{2} \right)}}{\pi \left(1 - u^{2}\right)} \sin{\left(u x \right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)}}{\pi \left(1 - u^{2}\right)} \cos{\left(u x \right)} 
cos(u*x)*(cos((u*pi)/2)/((pi*(1 - u^2)))) + sin(u*x)*((sin((u*pi)/2) - u)/((pi*(1 - u^2))))
Simplificación general [src] 
     /  /    pi\\             
- cos|u*|x - --|| + u*sin(u*x)
     \  \    2 //             
------------------------------
            /      2\         
         pi*\-1 + u /
usin⁡(ux)−cos⁡(u(x−π2))π(u2−1)\frac{u \sin{\left(u x \right)} - \cos{\left(u \left(x - \frac{\pi}{2}\right) \right)}}{\pi \left(u^{2} - 1\right)} 
(-cos(u*(x - pi/2)) + u*sin(u*x))/(pi*(-1 + u^2))
Respuesta numérica [src] 
(-u + sin((u*pi)/2))*sin(u*x)/(3.14159265358979 - 3.14159265358979*u^2) + cos(u*x)*cos((u*pi)/2)/(3.14159265358979 - 3.14159265358979*u^2)
(-u + sin((u*pi)/2))*sin(u*x)/(3.14159265358979 - 3.14159265358979*u^2) + cos(u*x)*cos((u*pi)/2)/(3.14159265358979 - 3.14159265358979*u^2)
Unión de expresiones racionales [src] 
/        /pi*u\\                        /pi*u\
|-u + sin|----||*sin(u*x) + cos(u*x)*cos|----|
\        \ 2  //                        \ 2  /
----------------------------------------------
                    /     2\                  
                 pi*\1 - u /
(−u+sin⁡(πu2))sin⁡(ux)+cos⁡(πu2)cos⁡(ux)π(1−u2)\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)} + \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} 
((-u + sin(pi*u/2))*sin(u*x) + cos(u*x)*cos(pi*u/2))/(pi*(1 - u^2))
Combinatoria [src] 
                         /pi*u\               /pi*u\
u*sin(u*x) - cos(u*x)*cos|----| - sin(u*x)*sin|----|
                         \ 2  /               \ 2  /
----------------------------------------------------
                pi*(1 + u)*(-1 + u)
usin⁡(ux)−sin⁡(πu2)sin⁡(ux)−cos⁡(πu2)cos⁡(ux)π(u−1)(u+1)\frac{u \sin{\left(u x \right)} - \sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)} - \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(u - 1\right) \left(u + 1\right)} 
(u*sin(u*x) - cos(u*x)*cos(pi*u/2) - sin(u*x)*sin(pi*u/2))/(pi*(1 + u)*(-1 + u))
Potencias [src] 
                   / pi*I*u    -pi*I*u \     /       /   -pi*I*u     pi*I*u\\                     
                   | ------    --------|     |       |   --------    ------||                     
/ I*u*x    -I*u*x\ |   2          2    |     |       |      2          2   ||                     
|e        e      | |e         e        |     |     I*\- e         + e      /| /   -I*u*x    I*u*x\
|------ + -------|*|------- + ---------|   I*|-u - -------------------------|*\- e       + e     /
\  2         2   / \   2          2    /     \                 2            /                     
---------------------------------------- - -------------------------------------------------------
                 /     2\                                            /     2\                     
              pi*\1 - u /                                       2*pi*\1 - u /
−i(−u−i(eiπu2−e−iπu2)2)(eiux−e−iux)2π(1−u2)+(eiπu22+e−iπu22)(eiux2+e−iux2)π(1−u2)- \frac{i \left(- u - \frac{i \left(e^{\frac{i \pi u}{2}} - e^{- \frac{i \pi u}{2}}\right)}{2}\right) \left(e^{i u x} - e^{- i u x}\right)}{2 \pi \left(1 - u^{2}\right)} + \frac{\left(\frac{e^{\frac{i \pi u}{2}}}{2} + \frac{e^{- \frac{i \pi u}{2}}}{2}\right) \left(\frac{e^{i u x}}{2} + \frac{e^{- i u x}}{2}\right)}{\pi \left(1 - u^{2}\right)} 
/        /pi*u\\                        /pi*u\
|-u + sin|----||*sin(u*x)   cos(u*x)*cos|----|
\        \ 2  //                        \ 2  /
------------------------- + ------------------
          /     2\                /     2\    
       pi*\1 - u /             pi*\1 - u /
(−u+sin⁡(πu2))sin⁡(ux)π(1−u2)+cos⁡(πu2)cos⁡(ux)π(1−u2)\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} 
(-u + sin(pi*u/2))*sin(u*x)/(pi*(1 - u^2)) + cos(u*x)*cos(pi*u/2)/(pi*(1 - u^2))
Denominador racional [src] 
                         /pi*u\               /pi*u\
u*sin(u*x) - cos(u*x)*cos|----| - sin(u*x)*sin|----|
                         \ 2  /               \ 2  /
----------------------------------------------------
                       /      2\                    
                    pi*\-1 + u /
usin⁡(ux)−sin⁡(πu2)sin⁡(ux)−cos⁡(πu2)cos⁡(ux)π(u2−1)\frac{u \sin{\left(u x \right)} - \sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)} - \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(u^{2} - 1\right)} 
(u*sin(u*x) - cos(u*x)*cos(pi*u/2) - sin(u*x)*sin(pi*u/2))/(pi*(-1 + u^2))
Compilar la expresión [src] 
/        /u*pi\\                        /u*pi\
|-u + sin|----||*sin(u*x)   cos(u*x)*cos|----|
\        \ 2  //                        \ 2  /
------------------------- + ------------------
          /     2\                /     2\    
       pi*\1 - u /             pi*\1 - u /
(−u+sin⁡(πu2))sin⁡(ux)π(1−u2)+cos⁡(πu2)cos⁡(ux)π(1−u2)\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} 
(-u + sin((u*pi)/2))*sin(u*x)/(pi*(1 - u^2)) + cos(u*x)*cos((u*pi)/2)/(pi*(1 - u^2))
Abrimos la expresión [src] 
            /pi*u\               /pi*u\             
cos(u*x)*cos|----|   sin(u*x)*sin|----|             
            \ 2  /               \ 2  /   u*sin(u*x)
------------------ + ------------------ - ----------
             2                    2                2
    pi - pi*u            pi - pi*u        pi - pi*u
−usin⁡(ux)−πu2+π+sin⁡(πu2)sin⁡(ux)−πu2+π+cos⁡(πu2)cos⁡(ux)−πu2+π- \frac{u \sin{\left(u x \right)}}{- \pi u^{2} + \pi} + \frac{\sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)}}{- \pi u^{2} + \pi} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{- \pi u^{2} + \pi} 
/   /u*pi\    \                        /u*pi\
|sin|----| - u|*sin(u*x)   cos(u*x)*cos|----|
\   \ 2  /    /                        \ 2  /
------------------------ + ------------------
         /     2\                /     2\    
      pi*\1 - u /             pi*\1 - u /
(−u+sin⁡(πu2))sin⁡(ux)π(1−u2)+cos⁡(πu2)cos⁡(ux)π(1−u2)\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} 
(sin((u*pi)/2) - u)*sin(u*x)/(pi*(1 - u^2)) + cos(u*x)*cos((u*pi)/2)/(pi*(1 - u^2))
Denominador común [src] 
                         /pi*u\               /pi*u\
u*sin(u*x) - cos(u*x)*cos|----| - sin(u*x)*sin|----|
                         \ 2  /               \ 2  /
----------------------------------------------------
                              2                     
                    -pi + pi*u
usin⁡(ux)−sin⁡(πu2)sin⁡(ux)−cos⁡(πu2)cos⁡(ux)πu2−π\frac{u \sin{\left(u x \right)} - \sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)} - \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi u^{2} - \pi} 
(u*sin(u*x) - cos(u*x)*cos(pi*u/2) - sin(u*x)*sin(pi*u/2))/(-pi + pi*u^2)
Parte trigonométrica [src] 
          /  /    pi\\                
          |u*|x - --||                
         2|  \    2 /|           /u*x\
  1 - tan |----------|    2*u*tan|---|
          \    2     /           \ 2 /
- -------------------- + -------------
          /  /    pi\\          2/u*x\
          |u*|x - --||   1 + tan |---|
         2|  \    2 /|           \ 2 /
  1 + tan |----------|                
          \    2     /                
--------------------------------------
                /      2\             
             pi*\-1 + u /
2utan⁡(ux2)tan⁡2(ux2)+1−1−tan⁡2(u(x−π2)2)tan⁡2(u(x−π2)2)+1π(u2−1)\frac{\frac{2 u \tan{\left(\frac{u x}{2} \right)}}{\tan^{2}{\left(\frac{u x}{2} \right)} + 1} - \frac{1 - \tan^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)}}{\tan^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)} + 1}}{\pi \left(u^{2} - 1\right)} 
/        /pi*u\\                        /pi*u\
|-u + sin|----||*sin(u*x)   cos(u*x)*cos|----|
\        \ 2  //                        \ 2  /
------------------------- + ------------------
          /     2\                /     2\    
       pi*\1 - u /             pi*\1 - u /
(−u+sin⁡(πu2))sin⁡(ux)π(1−u2)+cos⁡(πu2)cos⁡(ux)π(1−u2)\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} 
           /  /    pi\\                
           |u*|x - --||                
          2|  \    2 /|           /u*x\
  -1 + cot |----------|    2*u*cot|---|
           \    2     /           \ 2 /
- --------------------- + -------------
           /  /    pi\\          2/u*x\
           |u*|x - --||   1 + cot |---|
          2|  \    2 /|           \ 2 /
   1 + cot |----------|                
           \    2     /                
---------------------------------------
                 /      2\             
              pi*\-1 + u /
2ucot⁡(ux2)cot⁡2(ux2)+1−cot⁡2(u(x−π2)2)−1cot⁡2(u(x−π2)2)+1π(u2−1)\frac{\frac{2 u \cot{\left(\frac{u x}{2} \right)}}{\cot^{2}{\left(\frac{u x}{2} \right)} + 1} - \frac{\cot^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)} - 1}{\cot^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)} + 1}}{\pi \left(u^{2} - 1\right)} 
     /pi     /    pi\\             
- sin|-- + u*|x - --|| + u*sin(u*x)
     \2      \    2 //             
-----------------------------------
               /      2\           
            pi*\-1 + u /
usin⁡(ux)−sin⁡(u(x−π2)+π2)π(u2−1)\frac{u \sin{\left(u x \right)} - \sin{\left(u \left(x - \frac{\pi}{2}\right) + \frac{\pi}{2} \right)}}{\pi \left(u^{2} - 1\right)} 
         1                 u       
- --------------- + ---------------
     /  /    pi\\      /  pi      \
  sec|u*|x - --||   sec|- -- + u*x|
     \  \    2 //      \  2       /
-----------------------------------
               /      2\           
            pi*\-1 + u /
usec⁡(ux−π2)−1sec⁡(u(x−π2))π(u2−1)\frac{\frac{u}{\sec{\left(u x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(u \left(x - \frac{\pi}{2}\right) \right)}}}{\pi \left(u^{2} - 1\right)} 
/        /  pi   pi*u\\    /  pi      \               /pi*u\
|-u + cos|- -- + ----||*cos|- -- + u*x|   cos(u*x)*cos|----|
\        \  2     2  //    \  2       /               \ 2  /
--------------------------------------- + ------------------
                 /     2\                       /     2\    
              pi*\1 - u /                    pi*\1 - u /
(−u+cos⁡(πu2−π2))cos⁡(ux−π2)π(1−u2)+cos⁡(πu2)cos⁡(ux)π(1−u2)\frac{\left(- u + \cos{\left(\frac{\pi u}{2} - \frac{\pi}{2} \right)}\right) \cos{\left(u x - \frac{\pi}{2} \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} 
           1                u    
- -------------------- + --------
     /pi     /    pi\\   csc(u*x)
  csc|-- - u*|x - --||           
     \2      \    2 //           
---------------------------------
              /      2\          
           pi*\-1 + u /
ucsc⁡(ux)−1csc⁡(−u(x−π2)+π2)π(u2−1)\frac{\frac{u}{\csc{\left(u x \right)}} - \frac{1}{\csc{\left(- u \left(x - \frac{\pi}{2}\right) + \frac{\pi}{2} \right)}}}{\pi \left(u^{2} - 1\right)} 
                                            1               
                                     ---------------- - u   
                                        /  pi   pi*u\       
                                     sec|- -- + ----|       
              1                         \  2     2  /       
------------------------------ + ---------------------------
   /     2\             /pi*u\      /     2\    /  pi      \
pi*\1 - u /*sec(u*x)*sec|----|   pi*\1 - u /*sec|- -- + u*x|
                        \ 2  /                  \  2       /
−u+1sec⁡(πu2−π2)π(1−u2)sec⁡(ux−π2)+1π(1−u2)sec⁡(πu2)sec⁡(ux)\frac{- u + \frac{1}{\sec{\left(\frac{\pi u}{2} - \frac{\pi}{2} \right)}}}{\pi \left(1 - u^{2}\right) \sec{\left(u x - \frac{\pi}{2} \right)}} + \frac{1}{\pi \left(1 - u^{2}\right) \sec{\left(\frac{\pi u}{2} \right)} \sec{\left(u x \right)}} 
     /  /    pi\\             
- cos|u*|x - --|| + u*sin(u*x)
     \  \    2 //             
------------------------------
            /      2\         
         pi*\-1 + u /
usin⁡(ux)−cos⁡(u(x−π2))π(u2−1)\frac{u \sin{\left(u x \right)} - \cos{\left(u \left(x - \frac{\pi}{2}\right) \right)}}{\pi \left(u^{2} - 1\right)} 
  /           /pi*u\  \                                                        
  |      2*tan|----|  |                                                        
  |           \ 4  /  |    /u*x\                                               
2*|-u + --------------|*tan|---|                                               
  |            2/pi*u\|    \ 2 /         /       2/u*x\\ /       2/pi*u\\      
  |     1 + tan |----||                  |1 - tan |---||*|1 - tan |----||      
  \             \ 4  //                  \        \ 2 // \        \ 4  //      
-------------------------------- + --------------------------------------------
     /       2/u*x\\ /     2\         /       2/u*x\\ /       2/pi*u\\ /     2\
  pi*|1 + tan |---||*\1 - u /      pi*|1 + tan |---||*|1 + tan |----||*\1 - u /
     \        \ 2 //                  \        \ 2 // \        \ 4  //
(1−tan⁡2(πu4))(1−tan⁡2(ux2))π(1−u2)(tan⁡2(πu4)+1)(tan⁡2(ux2)+1)+2(−u+2tan⁡(πu4)tan⁡2(πu4)+1)tan⁡(ux2)π(1−u2)(tan⁡2(ux2)+1)\frac{\left(1 - \tan^{2}{\left(\frac{\pi u}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{u x}{2} \right)}\right)}{\pi \left(1 - u^{2}\right) \left(\tan^{2}{\left(\frac{\pi u}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{u x}{2} \right)} + 1\right)} + \frac{2 \left(- u + \frac{2 \tan{\left(\frac{\pi u}{4} \right)}}{\tan^{2}{\left(\frac{\pi u}{4} \right)} + 1}\right) \tan{\left(\frac{u x}{2} \right)}}{\pi \left(1 - u^{2}\right) \left(\tan^{2}{\left(\frac{u x}{2} \right)} + 1\right)} 
       1                                             
   --------- - u                                     
      /pi*u\                                         
   csc|----|                                         
      \ 2  /                         1               
-------------------- + ------------------------------
   /     2\               /     2\             /pi*u\
pi*\1 - u /*csc(u*x)   pi*\1 - u /*sec(u*x)*sec|----|
                                               \ 2  /
−u+1csc⁡(πu2)π(1−u2)csc⁡(ux)+1π(1−u2)sec⁡(πu2)sec⁡(ux)\frac{- u + \frac{1}{\csc{\left(\frac{\pi u}{2} \right)}}}{\pi \left(1 - u^{2}\right) \csc{\left(u x \right)}} + \frac{1}{\pi \left(1 - u^{2}\right) \sec{\left(\frac{\pi u}{2} \right)} \sec{\left(u x \right)}} 
/        /pi*u\\               /pi      \    /pi   pi*u\
|-u + sin|----||*sin(u*x)   sin|-- + u*x|*sin|-- + ----|
\        \ 2  //               \2       /    \2     2  /
------------------------- + ----------------------------
          /     2\                     /     2\         
       pi*\1 - u /                  pi*\1 - u /
(−u+sin⁡(πu2))sin⁡(ux)π(1−u2)+sin⁡(πu2+π2)sin⁡(ux+π2)π(1−u2)\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\sin{\left(\frac{\pi u}{2} + \frac{\pi}{2} \right)} \sin{\left(u x + \frac{\pi}{2} \right)}}{\pi \left(1 - u^{2}\right)} 
     /  /    pi\\        /  pi      \
- cos|u*|x - --|| + u*cos|- -- + u*x|
     \  \    2 //        \  2       /
-------------------------------------
                /      2\            
             pi*\-1 + u /
ucos⁡(ux−π2)−cos⁡(u(x−π2))π(u2−1)\frac{u \cos{\left(u x - \frac{\pi}{2} \right)} - \cos{\left(u \left(x - \frac{\pi}{2}\right) \right)}}{\pi \left(u^{2} - 1\right)} 
       1                                                       
   --------- - u                                               
      /pi*u\                                                   
   csc|----|                                                   
      \ 2  /                              1                    
-------------------- + ----------------------------------------
   /     2\               /     2\    /pi      \    /pi   pi*u\
pi*\1 - u /*csc(u*x)   pi*\1 - u /*csc|-- - u*x|*csc|-- - ----|
                                      \2       /    \2     2  /
−u+1csc⁡(πu2)π(1−u2)csc⁡(ux)+1π(1−u2)csc⁡(−πu2+π2)csc⁡(−ux+π2)\frac{- u + \frac{1}{\csc{\left(\frac{\pi u}{2} \right)}}}{\pi \left(1 - u^{2}\right) \csc{\left(u x \right)}} + \frac{1}{\pi \left(1 - u^{2}\right) \csc{\left(- \frac{\pi u}{2} + \frac{\pi}{2} \right)} \csc{\left(- u x + \frac{\pi}{2} \right)}} 
  /           /pi*u\  \                                                        
  |      2*cot|----|  |                                                        
  |           \ 4  /  |    /u*x\                                               
2*|-u + --------------|*cot|---|                                               
  |            2/pi*u\|    \ 2 /        /        2/u*x\\ /        2/pi*u\\     
  |     1 + cot |----||                 |-1 + cot |---||*|-1 + cot |----||     
  \             \ 4  //                 \         \ 2 // \         \ 4  //     
-------------------------------- + --------------------------------------------
     /       2/u*x\\ /     2\         /       2/u*x\\ /       2/pi*u\\ /     2\
  pi*|1 + cot |---||*\1 - u /      pi*|1 + cot |---||*|1 + cot |----||*\1 - u /
     \        \ 2 //                  \        \ 2 // \        \ 4  //
2(−u+2cot⁡(πu4)cot⁡2(πu4)+1)cot⁡(ux2)π(1−u2)(cot⁡2(ux2)+1)+(cot⁡2(πu4)−1)(cot⁡2(ux2)−1)π(1−u2)(cot⁡2(πu4)+1)(cot⁡2(ux2)+1)\frac{2 \left(- u + \frac{2 \cot{\left(\frac{\pi u}{4} \right)}}{\cot^{2}{\left(\frac{\pi u}{4} \right)} + 1}\right) \cot{\left(\frac{u x}{2} \right)}}{\pi \left(1 - u^{2}\right) \left(\cot^{2}{\left(\frac{u x}{2} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(\frac{\pi u}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{u x}{2} \right)} - 1\right)}{\pi \left(1 - u^{2}\right) \left(\cot^{2}{\left(\frac{\pi u}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{u x}{2} \right)} + 1\right)} 
2*(-u + 2*cot(pi*u/4)/(1 + cot(pi*u/4)^2))*cot(u*x/2)/(pi*(1 + cot(u*x/2)^2)*(1 - u^2)) + (-1 + cot(u*x/2)^2)*(-1 + cot(pi*u/4)^2)/(pi*(1 + cot(u*x/2)^2)*(1 + cot(pi*u/4)^2)*(1 - u^2))
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