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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(pi*x)/(2*pi)+sin(3*pi*x)/(6*pi)

¿Cómo vas a descomponer esta sin(pi*x)/(2*pi)+sin(3*pi*x)/(6*pi) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
sin(pi*x)   sin(3*pi*x)
--------- + -----------
   2*pi         6*pi
sin⁡(πx)2π+sin⁡(3πx)6π\frac{\sin{\left(\pi x \right)}}{2 \pi} + \frac{\sin{\left(3 \pi x \right)}}{6 \pi} 
sin(pi*x)/((2*pi)) + sin((3*pi)*x)/((6*pi))
Simplificación general [src] 
3*sin(pi*x) + sin(3*pi*x)
-------------------------
           6*pi
3sin⁡(πx)+sin⁡(3πx)6π\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi} 
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Respuesta numérica [src] 
0.0530516476972984*sin((3*pi)*x) + 0.159154943091895*sin(pi*x)
0.0530516476972984*sin((3*pi)*x) + 0.159154943091895*sin(pi*x)
Denominador racional [src] 
2*pi*sin(3*pi*x) + 6*pi*sin(pi*x)
---------------------------------
                   2             
              12*pi
6πsin⁡(πx)+2πsin⁡(3πx)12π2\frac{6 \pi \sin{\left(\pi x \right)} + 2 \pi \sin{\left(3 \pi x \right)}}{12 \pi^{2}} 
(2*pi*sin(3*pi*x) + 6*pi*sin(pi*x))/(12*pi^2)
Denominador común [src] 
3*sin(pi*x) + sin(3*pi*x)
-------------------------
           6*pi
3sin⁡(πx)+sin⁡(3πx)6π\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi} 
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Combinatoria [src] 
3*sin(pi*x) + sin(3*pi*x)
-------------------------
           6*pi
3sin⁡(πx)+sin⁡(3πx)6π\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi} 
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Potencias [src] 
    /   -pi*I*x    pi*I*x\     /   -3*pi*I*x    3*pi*I*x\
  I*\- e        + e      /   I*\- e          + e        /
- ------------------------ - ----------------------------
            4*pi                        12*pi
−i(eiπx−e−iπx)4π−i(e3iπx−e−3iπx)12π- \frac{i \left(e^{i \pi x} - e^{- i \pi x}\right)}{4 \pi} - \frac{i \left(e^{3 i \pi x} - e^{- 3 i \pi x}\right)}{12 \pi} 
-i*(-exp(-pi*i*x) + exp(pi*i*x))/(4*pi) - i*(-exp(-3*pi*i*x) + exp(3*pi*i*x))/(12*pi)
Unión de expresiones racionales [src] 
3*sin(pi*x) + sin(3*pi*x)
-------------------------
           6*pi
3sin⁡(πx)+sin⁡(3πx)6π\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi} 
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Parte trigonométrica [src] 
3*sin(pi*x) + sin(3*pi*x)
-------------------------
           6*pi
3sin⁡(πx)+sin⁡(3πx)6π\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi} 
        /pi*x\                 /3*pi*x\      
     tan|----|              tan|------|      
        \ 2  /                 \  2   /      
------------------- + -----------------------
   /       2/pi*x\\        /       2/3*pi*x\\
pi*|1 + tan |----||   3*pi*|1 + tan |------||
   \        \ 2  //        \        \  2   //
tan⁡(3πx2)3π(tan⁡2(3πx2)+1)+tan⁡(πx2)π(tan⁡2(πx2)+1)\frac{\tan{\left(\frac{3 \pi x}{2} \right)}}{3 \pi \left(\tan^{2}{\left(\frac{3 \pi x}{2} \right)} + 1\right)} + \frac{\tan{\left(\frac{\pi x}{2} \right)}}{\pi \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} 
      /3*pi*x\           /pi*x\  
 2*tan|------|      6*tan|----|  
      \  2   /           \ 2  /  
---------------- + --------------
       2/3*pi*x\          2/pi*x\
1 + tan |------|   1 + tan |----|
        \  2   /           \ 2  /
---------------------------------
               6*pi
2tan⁡(3πx2)tan⁡2(3πx2)+1+6tan⁡(πx2)tan⁡2(πx2)+16π\frac{\frac{2 \tan{\left(\frac{3 \pi x}{2} \right)}}{\tan^{2}{\left(\frac{3 \pi x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{\pi x}{2} \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{6 \pi} 
     /  pi       \      /  pi         \
3*cos|- -- + pi*x| + cos|- -- + 3*pi*x|
     \  2        /      \  2          /
---------------------------------------
                  6*pi
3cos⁡(πx−π2)+cos⁡(3πx−π2)6π\frac{3 \cos{\left(\pi x - \frac{\pi}{2} \right)} + \cos{\left(3 \pi x - \frac{\pi}{2} \right)}}{6 \pi} 
        /pi*x\                 /3*pi*x\      
     cot|----|              cot|------|      
        \ 2  /                 \  2   /      
------------------- + -----------------------
   /       2/pi*x\\        /       2/3*pi*x\\
pi*|1 + cot |----||   3*pi*|1 + cot |------||
   \        \ 2  //        \        \  2   //
cot⁡(3πx2)3π(cot⁡2(3πx2)+1)+cot⁡(πx2)π(cot⁡2(πx2)+1)\frac{\cot{\left(\frac{3 \pi x}{2} \right)}}{3 \pi \left(\cot^{2}{\left(\frac{3 \pi x}{2} \right)} + 1\right)} + \frac{\cot{\left(\frac{\pi x}{2} \right)}}{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} 
        1                   3        
------------------ + ----------------
   /  pi         \      /  pi       \
sec|- -- + 3*pi*x|   sec|- -- + pi*x|
   \  2          /      \  2        /
-------------------------------------
                 6*pi
1sec⁡(3πx−π2)+3sec⁡(πx−π2)6π\frac{\frac{1}{\sec{\left(3 \pi x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(\pi x - \frac{\pi}{2} \right)}}}{6 \pi} 
     1            3    
----------- + ---------
csc(3*pi*x)   csc(pi*x)
-----------------------
          6*pi
1csc⁡(3πx)+3csc⁡(πx)6π\frac{\frac{1}{\csc{\left(3 \pi x \right)}} + \frac{3}{\csc{\left(\pi x \right)}}}{6 \pi} 
          1                        1           
--------------------- + -----------------------
        /  pi       \           /  pi         \
2*pi*sec|- -- + pi*x|   6*pi*sec|- -- + 3*pi*x|
        \  2        /           \  2          /
16πsec⁡(3πx−π2)+12πsec⁡(πx−π2)\frac{1}{6 \pi \sec{\left(3 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{2 \pi \sec{\left(\pi x - \frac{\pi}{2} \right)}} 
      1                 1        
-------------- + ----------------
2*pi*csc(pi*x)   6*pi*csc(3*pi*x)
16πcsc⁡(3πx)+12πcsc⁡(πx)\frac{1}{6 \pi \csc{\left(3 \pi x \right)}} + \frac{1}{2 \pi \csc{\left(\pi x \right)}} 
   /  pi       \      /  pi         \
cos|- -- + pi*x|   cos|- -- + 3*pi*x|
   \  2        /      \  2          /
---------------- + ------------------
      2*pi                6*pi
cos⁡(πx−π2)2π+cos⁡(3πx−π2)6π\frac{\cos{\left(\pi x - \frac{\pi}{2} \right)}}{2 \pi} + \frac{\cos{\left(3 \pi x - \frac{\pi}{2} \right)}}{6 \pi} 
      /3*pi*x\           /pi*x\  
 2*cot|------|      6*cot|----|  
      \  2   /           \ 2  /  
---------------- + --------------
       2/3*pi*x\          2/pi*x\
1 + cot |------|   1 + cot |----|
        \  2   /           \ 2  /
---------------------------------
               6*pi
2cot⁡(3πx2)cot⁡2(3πx2)+1+6cot⁡(πx2)cot⁡2(πx2)+16π\frac{\frac{2 \cot{\left(\frac{3 \pi x}{2} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{\pi x}{2} \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{6 \pi} 
(2*cot(3*pi*x/2)/(1 + cot(3*pi*x/2)^2) + 6*cot(pi*x/2)/(1 + cot(pi*x/2)^2))/(6*pi)
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