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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(6*pi*x)/(12*pi)+sin(8*pi*x)/(16*pi)

¿Cómo vas a descomponer esta sin(6*pi*x)/(12*pi)+sin(8*pi*x)/(16*pi) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
sin(6*pi*x)   sin(8*pi*x)
----------- + -----------
   12*pi         16*pi
sin⁡(6πx)12π+sin⁡(8πx)16π\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{\sin{\left(8 \pi x \right)}}{16 \pi} 
sin((6*pi)*x)/((12*pi)) + sin((8*pi)*x)/((16*pi))
Simplificación general [src] 
sin(6*pi*x)   sin(8*pi*x)
----------- + -----------
     12            16    
-------------------------
            pi
sin⁡(6πx)12+sin⁡(8πx)16π\frac{\frac{\sin{\left(6 \pi x \right)}}{12} + \frac{\sin{\left(8 \pi x \right)}}{16}}{\pi} 
(sin(6*pi*x)/12 + sin(8*pi*x)/16)/pi
Respuesta numérica [src] 
0.0265258238486492*sin((6*pi)*x) + 0.0198943678864869*sin((8*pi)*x)
0.0265258238486492*sin((6*pi)*x) + 0.0198943678864869*sin((8*pi)*x)
Denominador racional [src] 
12*pi*sin(8*pi*x) + 16*pi*sin(6*pi*x)
-------------------------------------
                     2               
               192*pi
16πsin⁡(6πx)+12πsin⁡(8πx)192π2\frac{16 \pi \sin{\left(6 \pi x \right)} + 12 \pi \sin{\left(8 \pi x \right)}}{192 \pi^{2}} 
(12*pi*sin(8*pi*x) + 16*pi*sin(6*pi*x))/(192*pi^2)
Combinatoria [src] 
3*sin(8*pi*x) + 4*sin(6*pi*x)
-----------------------------
            48*pi
4sin⁡(6πx)+3sin⁡(8πx)48π\frac{4 \sin{\left(6 \pi x \right)} + 3 \sin{\left(8 \pi x \right)}}{48 \pi} 
(3*sin(8*pi*x) + 4*sin(6*pi*x))/(48*pi)
Potencias [src] 
    /   -6*pi*I*x    6*pi*I*x\     /   -8*pi*I*x    8*pi*I*x\
  I*\- e          + e        /   I*\- e          + e        /
- ---------------------------- - ----------------------------
             24*pi                          32*pi
−i(e6iπx−e−6iπx)24π−i(e8iπx−e−8iπx)32π- \frac{i \left(e^{6 i \pi x} - e^{- 6 i \pi x}\right)}{24 \pi} - \frac{i \left(e^{8 i \pi x} - e^{- 8 i \pi x}\right)}{32 \pi} 
-i*(-exp(-6*pi*i*x) + exp(6*pi*i*x))/(24*pi) - i*(-exp(-8*pi*i*x) + exp(8*pi*i*x))/(32*pi)
Denominador común [src] 
3*sin(8*pi*x) + 4*sin(6*pi*x)
-----------------------------
            48*pi
4sin⁡(6πx)+3sin⁡(8πx)48π\frac{4 \sin{\left(6 \pi x \right)} + 3 \sin{\left(8 \pi x \right)}}{48 \pi} 
(3*sin(8*pi*x) + 4*sin(6*pi*x))/(48*pi)
Unión de expresiones racionales [src] 
3*sin(8*pi*x) + 4*sin(6*pi*x)
-----------------------------
            48*pi
4sin⁡(6πx)+3sin⁡(8πx)48π\frac{4 \sin{\left(6 \pi x \right)} + 3 \sin{\left(8 \pi x \right)}}{48 \pi} 
(3*sin(8*pi*x) + 4*sin(6*pi*x))/(48*pi)
Parte trigonométrica [src] 
      tan(3*pi*x)               tan(4*pi*x)      
----------------------- + -----------------------
     /       2        \        /       2        \
6*pi*\1 + tan (3*pi*x)/   8*pi*\1 + tan (4*pi*x)/
tan⁡(4πx)8π(tan⁡2(4πx)+1)+tan⁡(3πx)6π(tan⁡2(3πx)+1)\frac{\tan{\left(4 \pi x \right)}}{8 \pi \left(\tan^{2}{\left(4 \pi x \right)} + 1\right)} + \frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)} 
sin(6*pi*x)             1           
----------- + ----------------------
   12*pi      16*pi*csc(pi - 8*pi*x)
sin⁡(6πx)12π+116πcsc⁡(−8πx+π)\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \csc{\left(- 8 \pi x + \pi \right)}} 
                 2                 /       2/pi         \\
              sin (pi*(1/4 + 4*x))*|1 - cot |-- + 4*pi*x||
sin(6*pi*x)                        \        \4          //
----------- + --------------------------------------------
   12*pi                         16*pi
(1−cot⁡2(4πx+π4))sin⁡2(π(4x+14))16π+sin⁡(6πx)12π\frac{\left(1 - \cot^{2}{\left(4 \pi x + \frac{\pi}{4} \right)}\right) \sin^{2}{\left(\pi \left(4 x + \frac{1}{4}\right) \right)}}{16 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
sin(6*pi*x)   cos(4*pi*x)*sin(4*pi*x)
----------- + -----------------------
   12*pi                8*pi
sin⁡(4πx)cos⁡(4πx)8π+sin⁡(6πx)12π\frac{\sin{\left(4 \pi x \right)} \cos{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
2*sin(6*pi*x) + 3*cos(4*pi*x)*sin(4*pi*x)
-----------------------------------------
                  24*pi
3sin⁡(4πx)cos⁡(4πx)+2sin⁡(6πx)24π\frac{3 \sin{\left(4 \pi x \right)} \cos{\left(4 \pi x \right)} + 2 \sin{\left(6 \pi x \right)}}{24 \pi} 
        1                           1                
----------------- + ---------------------------------
12*pi*csc(6*pi*x)                       /pi         \
                    8*pi*csc(4*pi*x)*csc|-- - 4*pi*x|
                                        \2          /
112πcsc⁡(6πx)+18πcsc⁡(4πx)csc⁡(−4πx+π2)\frac{1}{12 \pi \csc{\left(6 \pi x \right)}} + \frac{1}{8 \pi \csc{\left(4 \pi x \right)} \csc{\left(- 4 \pi x + \frac{\pi}{2} \right)}} 
           1                       1        
------------------------ + -----------------
         /  pi         \   16*pi*csc(8*pi*x)
12*pi*sec|- -- + 6*pi*x|                    
         \  2          /
112πsec⁡(6πx−π2)+116πcsc⁡(8πx)\frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{16 \pi \csc{\left(8 \pi x \right)}} 
                                      2/pi         \   
                              -1 + tan |-- + 4*pi*x|   
      cot(3*pi*x)                      \4          /   
----------------------- + -----------------------------
     /       2        \         /       2/pi         \\
6*pi*\1 + cot (3*pi*x)/   16*pi*|1 + tan |-- + 4*pi*x||
                                \        \4          //
tan⁡2(4πx+π4)−116π(tan⁡2(4πx+π4)+1)+cot⁡(3πx)6π(cot⁡2(3πx)+1)\frac{\tan^{2}{\left(4 \pi x + \frac{\pi}{4} \right)} - 1}{16 \pi \left(\tan^{2}{\left(4 \pi x + \frac{\pi}{4} \right)} + 1\right)} + \frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)} 
      cot(3*pi*x)               cot(4*pi*x)      
----------------------- + -----------------------
     /       2        \        /       2        \
6*pi*\1 + cot (3*pi*x)/   8*pi*\1 + cot (4*pi*x)/
cot⁡(4πx)8π(cot⁡2(4πx)+1)+cot⁡(3πx)6π(cot⁡2(3πx)+1)\frac{\cot{\left(4 \pi x \right)}}{8 \pi \left(\cot^{2}{\left(4 \pi x \right)} + 1\right)} + \frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)} 
                             2                    
      cot(3*pi*x)         cot (2*pi*x)*cot(4*pi*x)
----------------------- + ------------------------
     /       2        \                          2
6*pi*\1 + cot (3*pi*x)/        /       2        \ 
                          2*pi*\1 + cot (2*pi*x)/
cot⁡(3πx)6π(cot⁡2(3πx)+1)+cot⁡2(2πx)cot⁡(4πx)2π(cot⁡2(2πx)+1)2\frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)} + \frac{\cot^{2}{\left(2 \pi x \right)} \cot{\left(4 \pi x \right)}}{2 \pi \left(\cot^{2}{\left(2 \pi x \right)} + 1\right)^{2}} 
           1                         1           
------------------------ + ----------------------
         /  pi         \            /pi         \
12*pi*sec|- -- + 6*pi*x|   16*pi*sec|-- - 8*pi*x|
         \  2          /            \2          /
112πsec⁡(6πx−π2)+116πsec⁡(−8πx+π2)\frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{16 \pi \sec{\left(- 8 \pi x + \frac{\pi}{2} \right)}} 
           1                          1            
------------------------ + ------------------------
         /  pi         \            /  pi         \
12*pi*sec|- -- + 6*pi*x|   16*pi*sec|- -- + 8*pi*x|
         \  2          /            \  2          /
116πsec⁡(8πx−π2)+112πsec⁡(6πx−π2)\frac{1}{16 \pi \sec{\left(8 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} 
        1                     1           
----------------- + ----------------------
12*pi*csc(6*pi*x)   16*pi*csc(pi - 8*pi*x)
116πcsc⁡(−8πx+π)+112πcsc⁡(6πx)\frac{1}{16 \pi \csc{\left(- 8 \pi x + \pi \right)}} + \frac{1}{12 \pi \csc{\left(6 \pi x \right)}} 
                 2                    
sin(6*pi*x)   cos (4*pi*x)*tan(4*pi*x)
----------- + ------------------------
   12*pi                8*pi
sin⁡(6πx)12π+cos⁡2(4πx)tan⁡(4πx)8π\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{\cos^{2}{\left(4 \pi x \right)} \tan{\left(4 \pi x \right)}}{8 \pi} 
sin(6*pi*x)     sin(4*pi*x)   
----------- + ----------------
   12*pi      8*pi*sec(4*pi*x)
sin⁡(4πx)8πsec⁡(4πx)+sin⁡(6πx)12π\frac{\sin{\left(4 \pi x \right)}}{8 \pi \sec{\left(4 \pi x \right)}} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
                 2                    
sin(6*pi*x)   sin (4*pi*x)*cot(4*pi*x)
----------- + ------------------------
   12*pi                8*pi
sin⁡2(4πx)cot⁡(4πx)8π+sin⁡(6πx)12π\frac{\sin^{2}{\left(4 \pi x \right)} \cot{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
   /  pi         \                  /  pi         \
cos|- -- + 6*pi*x|   cos(4*pi*x)*cos|- -- + 4*pi*x|
   \  2          /                  \  2          /
------------------ + ------------------------------
      12*pi                       8*pi
cos⁡(4πx)cos⁡(4πx−π2)8π+cos⁡(6πx−π2)12π\frac{\cos{\left(4 \pi x \right)} \cos{\left(4 \pi x - \frac{\pi}{2} \right)}}{8 \pi} + \frac{\cos{\left(6 \pi x - \frac{\pi}{2} \right)}}{12 \pi} 
                                            2            
                          /       2        \             
      tan(3*pi*x)         \1 - tan (2*pi*x)/ *tan(4*pi*x)
----------------------- + -------------------------------
     /       2        \                              2   
6*pi*\1 + tan (3*pi*x)/            /       2        \    
                              8*pi*\1 + tan (2*pi*x)/
(1−tan⁡2(2πx))2tan⁡(4πx)8π(tan⁡2(2πx)+1)2+tan⁡(3πx)6π(tan⁡2(3πx)+1)\frac{\left(1 - \tan^{2}{\left(2 \pi x \right)}\right)^{2} \tan{\left(4 \pi x \right)}}{8 \pi \left(\tan^{2}{\left(2 \pi x \right)} + 1\right)^{2}} + \frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)} 
                                                     2        
                                  /        2        \         
      cot(3*pi*x)                 \-1 + cot (2*pi*x)/         
----------------------- + ------------------------------------
     /       2        \                          2            
6*pi*\1 + cot (3*pi*x)/        /       2        \             
                          8*pi*\1 + cot (2*pi*x)/ *cot(4*pi*x)
(cot⁡2(2πx)−1)28π(cot⁡2(2πx)+1)2cot⁡(4πx)+cot⁡(3πx)6π(cot⁡2(3πx)+1)\frac{\left(\cot^{2}{\left(2 \pi x \right)} - 1\right)^{2}}{8 \pi \left(\cot^{2}{\left(2 \pi x \right)} + 1\right)^{2} \cot{\left(4 \pi x \right)}} + \frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)} 
   /  pi         \              
cos|- -- + 6*pi*x|              
   \  2          /   sin(8*pi*x)
------------------ + -----------
      12*pi             16*pi
sin⁡(8πx)16π+cos⁡(6πx−π2)12π\frac{\sin{\left(8 \pi x \right)}}{16 \pi} + \frac{\cos{\left(6 \pi x - \frac{\pi}{2} \right)}}{12 \pi} 
                                2                         
              /       2        \     4                    
sin(6*pi*x)   \1 - tan (2*pi*x)/ *cos (2*pi*x)*tan(4*pi*x)
----------- + --------------------------------------------
   12*pi                          8*pi
(1−tan⁡2(2πx))2cos⁡4(2πx)tan⁡(4πx)8π+sin⁡(6πx)12π\frac{\left(1 - \tan^{2}{\left(2 \pi x \right)}\right)^{2} \cos^{4}{\left(2 \pi x \right)} \tan{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
                                         2                    
      tan(3*pi*x)                     tan (2*pi*x)            
----------------------- + ------------------------------------
     /       2        \                          2            
6*pi*\1 + tan (3*pi*x)/        /       2        \             
                          2*pi*\1 + tan (2*pi*x)/ *tan(4*pi*x)
tan⁡(3πx)6π(tan⁡2(3πx)+1)+tan⁡2(2πx)2π(tan⁡2(2πx)+1)2tan⁡(4πx)\frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)} + \frac{\tan^{2}{\left(2 \pi x \right)}}{2 \pi \left(\tan^{2}{\left(2 \pi x \right)} + 1\right)^{2} \tan{\left(4 \pi x \right)}} 
                 /  pi         \
              cos|- -- + 8*pi*x|
sin(6*pi*x)      \  2          /
----------- + ------------------
   12*pi            16*pi
sin⁡(6πx)12π+cos⁡(8πx−π2)16π\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{\cos{\left(8 \pi x - \frac{\pi}{2} \right)}}{16 \pi} 
sin(6*pi*x)             1           
----------- + ----------------------
   12*pi               /pi         \
              16*pi*sec|-- - 8*pi*x|
                       \2          /
sin⁡(6πx)12π+116πsec⁡(−8πx+π2)\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \sec{\left(- 8 \pi x + \frac{\pi}{2} \right)}} 
sin(6*pi*x)           1        
----------- + -----------------
   12*pi      16*pi*csc(8*pi*x)
sin⁡(6πx)12π+116πcsc⁡(8πx)\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \csc{\left(8 \pi x \right)}} 
sin(6*pi*x)              1            
----------- + ------------------------
   12*pi               /  pi         \
              16*pi*sec|- -- + 8*pi*x|
                       \  2          /
sin⁡(6πx)12π+116πsec⁡(8πx−π2)\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \sec{\left(8 \pi x - \frac{\pi}{2} \right)}} 
                 2            2/pi         \
              sin (4*pi*x)*sin |-- + 4*pi*x|
sin(6*pi*x)                    \2          /
----------- + ------------------------------
   12*pi             4*pi*sin(8*pi*x)
sin⁡2(4πx)sin⁡2(4πx+π2)4πsin⁡(8πx)+sin⁡(6πx)12π\frac{\sin^{2}{\left(4 \pi x \right)} \sin^{2}{\left(4 \pi x + \frac{\pi}{2} \right)}}{4 \pi \sin{\left(8 \pi x \right)}} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
           1                                1                 
------------------------ + -----------------------------------
         /  pi         \                       /  pi         \
12*pi*sec|- -- + 6*pi*x|   8*pi*sec(4*pi*x)*sec|- -- + 4*pi*x|
         \  2          /                       \  2          /
112πsec⁡(6πx−π2)+18πsec⁡(4πx)sec⁡(4πx−π2)\frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{8 \pi \sec{\left(4 \pi x \right)} \sec{\left(4 \pi x - \frac{\pi}{2} \right)}} 
                                 2                         
              /        2        \     4                    
sin(6*pi*x)   \-1 + cot (2*pi*x)/ *sin (2*pi*x)*tan(4*pi*x)
----------- + ---------------------------------------------
   12*pi                           8*pi
(cot⁡2(2πx)−1)2sin⁡4(2πx)tan⁡(4πx)8π+sin⁡(6πx)12π\frac{\left(\cot^{2}{\left(2 \pi x \right)} - 1\right)^{2} \sin^{4}{\left(2 \pi x \right)} \tan{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
   /  pi         \      /  pi         \
cos|- -- + 6*pi*x|   cos|- -- + 8*pi*x|
   \  2          /      \  2          /
------------------ + ------------------
      12*pi                16*pi
cos⁡(6πx−π2)12π+cos⁡(8πx−π2)16π\frac{\cos{\left(6 \pi x - \frac{\pi}{2} \right)}}{12 \pi} + \frac{\cos{\left(8 \pi x - \frac{\pi}{2} \right)}}{16 \pi} 
                                     2/pi         \    
                              1 - cot |-- + 4*pi*x|    
      tan(3*pi*x)                     \4          /    
----------------------- + -----------------------------
     /       2        \         /       2/pi         \\
6*pi*\1 + tan (3*pi*x)/   16*pi*|1 + cot |-- + 4*pi*x||
                                \        \4          //
1−cot⁡2(4πx+π4)16π(cot⁡2(4πx+π4)+1)+tan⁡(3πx)6π(tan⁡2(3πx)+1)\frac{1 - \cot^{2}{\left(4 \pi x + \frac{\pi}{4} \right)}}{16 \pi \left(\cot^{2}{\left(4 \pi x + \frac{\pi}{4} \right)} + 1\right)} + \frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)} 
        1                   1        
----------------- + -----------------
12*pi*csc(6*pi*x)   16*pi*csc(8*pi*x)
116πcsc⁡(8πx)+112πcsc⁡(6πx)\frac{1}{16 \pi \csc{\left(8 \pi x \right)}} + \frac{1}{12 \pi \csc{\left(6 \pi x \right)}} 
sin(6*pi*x)        sin(4*pi*x)     
----------- + ---------------------
   12*pi              /pi         \
              8*pi*csc|-- - 4*pi*x|
                      \2          /
sin⁡(4πx)8πcsc⁡(−4πx+π2)+sin⁡(6πx)12π\frac{\sin{\left(4 \pi x \right)}}{8 \pi \csc{\left(- 4 \pi x + \frac{\pi}{2} \right)}} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi} 
sin(6*pi*x)/(12*pi) + sin(4*pi*x)/(8*pi*csc(pi/2 - 4*pi*x))
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