Sr Examen

Otras calculadoras

Simplificación de expresiones/ Descomposición en fracciones simples/ ((-cos(pi*x/(3+k)))/(2*pi/(3+k))-3/(2*(k-pi))*cos((k-pi)*x/3)+cos(k*x)/k)/3

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    / pi*x\                                         
-cos|-----|                                         
    \3 + k/        3         /(k - pi)*x\   cos(k*x)
------------ - ----------*cos|----------| + --------
  / 2*pi\      2*(k - pi)    \    3     /      k    
  |-----|                                           
  \3 + k/                                           
----------------------------------------------------
                         3
(−32(k−π)cos⁡(x(k−π)3)+(−1)cos⁡(πxk+3)2π1k+3)+cos⁡(kx)k3\frac{\left(- \frac{3}{2 \left(k - \pi\right)} \cos{\left(\frac{x \left(k - \pi\right)}{3} \right)} + \frac{\left(-1\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{2 \pi \frac{1}{k + 3}}\right) + \frac{\cos{\left(k x \right)}}{k}}{3} 
((-cos((pi*x)/(3 + k)))/(((2*pi)/(3 + k))) - 3/((2*(k - pi)))*cos(((k - pi)*x)/3) + cos(k*x)/k)/3
Simplificación general [src] 
            /x*(k - pi)\                                                  / pi*x\
- 3*pi*k*cos|----------| + 2*pi*(k - pi)*cos(k*x) - k*(3 + k)*(k - pi)*cos|-----|
            \    3     /                                                  \3 + k/
---------------------------------------------------------------------------------
                                 6*pi*k*(k - pi)
−k(k+3)(k−π)cos⁡(πxk+3)−3πkcos⁡(x(k−π)3)+2π(k−π)cos⁡(kx)6πk(k−π)\frac{- k \left(k + 3\right) \left(k - \pi\right) \cos{\left(\frac{\pi x}{k + 3} \right)} - 3 \pi k \cos{\left(\frac{x \left(k - \pi\right)}{3} \right)} + 2 \pi \left(k - \pi\right) \cos{\left(k x \right)}}{6 \pi k \left(k - \pi\right)} 
(-3*pi*k*cos(x*(k - pi)/3) + 2*pi*(k - pi)*cos(k*x) - k*(3 + k)*(k - pi)*cos(pi*x/(3 + k)))/(6*pi*k*(k - pi))
Potencias [src] 
     /x*(k - pi)\                         / pi*x\
  cos|----------|              (3 + k)*cos|-----|
     \    3     /   cos(k*x)              \3 + k/
- --------------- + -------- - ------------------
    -2*pi + 2*k       3*k             6*pi
−(k+3)cos⁡(πxk+3)6π−cos⁡(x(k−π)3)2k−2π+cos⁡(kx)3k- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k} 
     /x*(k - pi)\              /3   k\    / pi*x\
  cos|----------|              |- + -|*cos|-----|
     \    3     /   cos(k*x)   \2   2/    \3 + k/
- --------------- + -------- - ------------------
    -2*pi + 2*k       3*k             3*pi
−(k2+32)cos⁡(πxk+3)3π−cos⁡(x(k−π)3)2k−2π+cos⁡(kx)3k- \frac{\left(\frac{k}{2} + \frac{3}{2}\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{3 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k} 
   -I*x*(k - pi)     I*x*(k - pi)                              /   pi*I*x    -pi*I*x \
   --------------    ------------                              |   ------    --------|
         3                3          I*k*x    -I*k*x           |   3 + k      3 + k  |
  e                 e               e        e                 |  e         e        |
  --------------- + -------------   ------ + -------   (3 + k)*|- ------- - ---------|
         2                2           2         2              \     2          2    /
- ------------------------------- + ---------------- + -------------------------------
            -2*pi + 2*k                   3*k                        6*pi
(k+3)(−eiπxk+32−e−iπxk+32)6π−eix(k−π)32+e−ix(k−π)322k−2π+eikx2+e−ikx23k\frac{\left(k + 3\right) \left(- \frac{e^{\frac{i \pi x}{k + 3}}}{2} - \frac{e^{- \frac{i \pi x}{k + 3}}}{2}\right)}{6 \pi} - \frac{\frac{e^{\frac{i x \left(k - \pi\right)}{3}}}{2} + \frac{e^{- \frac{i x \left(k - \pi\right)}{3}}}{2}}{2 k - 2 \pi} + \frac{\frac{e^{i k x}}{2} + \frac{e^{- i k x}}{2}}{3 k} 
-(exp(-i*x*(k - pi)/3)/2 + exp(i*x*(k - pi)/3)/2)/(-2*pi + 2*k) + (exp(i*k*x)/2 + exp(-i*k*x)/2)/(3*k) + (3 + k)*(-exp(pi*i*x/(3 + k))/2 - exp(-pi*i*x/(3 + k))/2)/(6*pi)
Unión de expresiones racionales [src] 
  /          /x*(k - pi)\                       / pi*x\\                         
k*|- 3*pi*cos|----------| - (3 + k)*(k - pi)*cos|-----|| + 2*pi*(k - pi)*cos(k*x)
  \          \    3     /                       \3 + k//                         
---------------------------------------------------------------------------------
                                 6*pi*k*(k - pi)
k(−(k+3)(k−π)cos⁡(πxk+3)−3πcos⁡(x(k−π)3))+2π(k−π)cos⁡(kx)6πk(k−π)\frac{k \left(- \left(k + 3\right) \left(k - \pi\right) \cos{\left(\frac{\pi x}{k + 3} \right)} - 3 \pi \cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}\right) + 2 \pi \left(k - \pi\right) \cos{\left(k x \right)}}{6 \pi k \left(k - \pi\right)} 
(k*(-3*pi*cos(x*(k - pi)/3) - (3 + k)*(k - pi)*cos(pi*x/(3 + k))) + 2*pi*(k - pi)*cos(k*x))/(6*pi*k*(k - pi))
Denominador común [src] 
   / pi*x\    3    / pi*x\       2               2    / pi*x\       2    / pi*x\             / pi*x\                               /  pi*x   k*x\
cos|-----|   k *cos|-----| + 2*pi *cos(k*x) + 3*k *cos|-----| - k*pi *cos|-----| - 3*pi*k*cos|-----| - 2*pi*k*cos(k*x) + 3*pi*k*cos|- ---- + ---|
   \3 + k/         \3 + k/                            \3 + k/            \3 + k/             \3 + k/                               \   3      3 /
---------- - ------------------------------------------------------------------------------------------------------------------------------------
    6                                                                        2         2                                                         
                                                                     - 6*k*pi  + 6*pi*k
cos⁡(πxk+3)6−k3cos⁡(πxk+3)+3k2cos⁡(πxk+3)−2πkcos⁡(kx)−π2kcos⁡(πxk+3)−3πkcos⁡(πxk+3)+3πkcos⁡(kx3−πx3)+2π2cos⁡(kx)6πk2−6π2k\frac{\cos{\left(\frac{\pi x}{k + 3} \right)}}{6} - \frac{k^{3} \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} - 2 \pi k \cos{\left(k x \right)} - \pi^{2} k \cos{\left(\frac{\pi x}{k + 3} \right)} - 3 \pi k \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 \pi k \cos{\left(\frac{k x}{3} - \frac{\pi x}{3} \right)} + 2 \pi^{2} \cos{\left(k x \right)}}{6 \pi k^{2} - 6 \pi^{2} k} 
cos(pi*x/(3 + k))/6 - (k^3*cos(pi*x/(3 + k)) + 2*pi^2*cos(k*x) + 3*k^2*cos(pi*x/(3 + k)) - k*pi^2*cos(pi*x/(3 + k)) - 3*pi*k*cos(pi*x/(3 + k)) - 2*pi*k*cos(k*x) + 3*pi*k*cos(-pi*x/3 + k*x/3))/(-6*k*pi^2 + 6*pi*k^2)
Denominador racional [src] 
     2    / pi*x\       2               3    / pi*x\             /  pi*x   k*x\         2    / pi*x\                               / pi*x\
- 6*k *cos|-----| - 4*pi *cos(k*x) - 2*k *cos|-----| - 6*pi*k*cos|- ---- + ---| + 2*pi*k *cos|-----| + 4*pi*k*cos(k*x) + 6*pi*k*cos|-----|
          \3 + k/                            \3 + k/             \   3      3 /              \3 + k/                               \3 + k/
------------------------------------------------------------------------------------------------------------------------------------------
                                                           6*pi*k*(-2*pi + 2*k)
−2k3cos⁡(πxk+3)−6k2cos⁡(πxk+3)+2πk2cos⁡(πxk+3)+4πkcos⁡(kx)+6πkcos⁡(πxk+3)−6πkcos⁡(kx3−πx3)−4π2cos⁡(kx)6πk(2k−2π)\frac{- 2 k^{3} \cos{\left(\frac{\pi x}{k + 3} \right)} - 6 k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} + 2 \pi k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} + 4 \pi k \cos{\left(k x \right)} + 6 \pi k \cos{\left(\frac{\pi x}{k + 3} \right)} - 6 \pi k \cos{\left(\frac{k x}{3} - \frac{\pi x}{3} \right)} - 4 \pi^{2} \cos{\left(k x \right)}}{6 \pi k \left(2 k - 2 \pi\right)} 
(-6*k^2*cos(pi*x/(3 + k)) - 4*pi^2*cos(k*x) - 2*k^3*cos(pi*x/(3 + k)) - 6*pi*k*cos(-pi*x/3 + k*x/3) + 2*pi*k^2*cos(pi*x/(3 + k)) + 4*pi*k*cos(k*x) + 6*pi*k*cos(pi*x/(3 + k)))/(6*pi*k*(-2*pi + 2*k))
Abrimos la expresión [src] 
     / pi*x\                 /pi*x\    /k*x\      /pi*x\    /k*x\        / pi*x\
  cos|-----|              cos|----|*cos|---|   sin|----|*sin|---|   k*cos|-----|
     \3 + k/   cos(k*x)      \ 3  /    \ 3 /      \ 3  /    \ 3 /        \3 + k/
- ---------- + -------- - ------------------ - ------------------ - ------------
     2*pi        3*k         -2*pi + 2*k          -2*pi + 2*k           6*pi
−kcos⁡(πxk+3)6π−cos⁡(πxk+3)2π−sin⁡(πx3)sin⁡(kx3)2k−2π−cos⁡(πx3)cos⁡(kx3)2k−2π+cos⁡(kx)3k- \frac{k \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{\pi x}{k + 3} \right)}}{2 \pi} - \frac{\sin{\left(\frac{\pi x}{3} \right)} \sin{\left(\frac{k x}{3} \right)}}{2 k - 2 \pi} - \frac{\cos{\left(\frac{\pi x}{3} \right)} \cos{\left(\frac{k x}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k} 
     /(k - pi)*x\                         / pi*x\
  cos|----------|              (3 + k)*cos|-----|
     \    3     /   cos(k*x)              \3 + k/
- --------------- + -------- - ------------------
     2*(k - pi)       3*k             6*pi
−(k+3)cos⁡(πxk+3)6π−cos⁡(x(k−π)3)2(k−π)+cos⁡(kx)3k- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 \left(k - \pi\right)} + \frac{\cos{\left(k x \right)}}{3 k} 
-cos(((k - pi)*x)/3)/(2*(k - pi)) + cos(k*x)/(3*k) - (3 + k)*cos((pi*x)/(3 + k))/(6*pi)
Compilar la expresión [src] 
     /(k - pi)*x\                         / pi*x\
  cos|----------|              (3 + k)*cos|-----|
     \    3     /   cos(k*x)              \3 + k/
- --------------- + -------- - ------------------
    -2*pi + 2*k       3*k             6*pi
−(k+3)cos⁡(πxk+3)6π−cos⁡(x(k−π)3)2k−2π+cos⁡(kx)3k- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k} 
-cos(((k - pi)*x)/3)/(-2*pi + 2*k) + cos(k*x)/(3*k) - (3 + k)*cos((pi*x)/(3 + k))/(6*pi)
Combinatoria [src] 
 / 3    / pi*x\       2               2    / pi*x\       2    / pi*x\             / pi*x\                               /  pi*x   k*x\\ 
-|k *cos|-----| + 2*pi *cos(k*x) + 3*k *cos|-----| - pi*k *cos|-----| - 3*pi*k*cos|-----| - 2*pi*k*cos(k*x) + 3*pi*k*cos|- ---- + ---|| 
 \      \3 + k/                            \3 + k/            \3 + k/             \3 + k/                               \   3      3 // 
----------------------------------------------------------------------------------------------------------------------------------------
                                                            6*pi*k*(k - pi)
−k3cos⁡(πxk+3)−πk2cos⁡(πxk+3)+3k2cos⁡(πxk+3)−2πkcos⁡(kx)−3πkcos⁡(πxk+3)+3πkcos⁡(kx3−πx3)+2π2cos⁡(kx)6πk(k−π)- \frac{k^{3} \cos{\left(\frac{\pi x}{k + 3} \right)} - \pi k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} - 2 \pi k \cos{\left(k x \right)} - 3 \pi k \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 \pi k \cos{\left(\frac{k x}{3} - \frac{\pi x}{3} \right)} + 2 \pi^{2} \cos{\left(k x \right)}}{6 \pi k \left(k - \pi\right)} 
-(k^3*cos(pi*x/(3 + k)) + 2*pi^2*cos(k*x) + 3*k^2*cos(pi*x/(3 + k)) - pi*k^2*cos(pi*x/(3 + k)) - 3*pi*k*cos(pi*x/(3 + k)) - 2*pi*k*cos(k*x) + 3*pi*k*cos(-pi*x/3 + k*x/3))/(6*pi*k*(k - pi))
Respuesta numérica [src] 
0.333333333333333*cos(k*x)/k - 1.0*cos(((k - pi)*x)/3)/(-6.28318530717959 + 2.0*k) - 0.333333333333333*(0.477464829275686 + 0.159154943091895*k)*cos((pi*x)/(3 + k))
0.333333333333333*cos(k*x)/k - 1.0*cos(((k - pi)*x)/3)/(-6.28318530717959 + 2.0*k) - 0.333333333333333*(0.477464829275686 + 0.159154943091895*k)*cos((pi*x)/(3 + k))
Parte trigonométrica [src] 
     /x*(k - pi)\                         / pi*x\
  cos|----------|              (3 + k)*cos|-----|
     \    3     /   cos(k*x)              \3 + k/
- --------------- + -------- - ------------------
    -2*pi + 2*k       3*k             6*pi
−(k+3)cos⁡(πxk+3)6π−cos⁡(x(k−π)3)2k−2π+cos⁡(kx)3k- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k} 
     /pi   x*(k - pi)\      /pi      \              /pi    pi*x\
  sin|-- + ----------|   sin|-- + k*x|   (3 + k)*sin|-- + -----|
     \2        3     /      \2       /              \2    3 + k/
- -------------------- + ------------- - -----------------------
      -2*pi + 2*k             3*k                  6*pi
−(k+3)sin⁡(πxk+3+π2)6π−sin⁡(x(k−π)3+π2)2k−2π+sin⁡(kx+π2)3k- \frac{\left(k + 3\right) \sin{\left(\frac{\pi x}{k + 3} + \frac{\pi}{2} \right)}}{6 \pi} - \frac{\sin{\left(\frac{x \left(k - \pi\right)}{3} + \frac{\pi}{2} \right)}}{2 k - 2 \pi} + \frac{\sin{\left(k x + \frac{\pi}{2} \right)}}{3 k} 
     /x*(k - pi)\                         / pi*x\
  cos|----------|              (3 + k)*cos|-----|
     \    3     /   cos(k*x)              \3 + k/
- --------------- + -------- - ------------------
     2*(k - pi)       3*k             6*pi
−(k+3)cos⁡(πxk+3)6π−cos⁡(x(k−π)3)2(k−π)+cos⁡(kx)3k- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 \left(k - \pi\right)} + \frac{\cos{\left(k x \right)}}{3 k} 
                 2/x*(k - pi)\                     2/k*x\      /       2/   pi*x  \\        
          1 - tan |----------|              1 - tan |---|      |1 - tan |---------||*(3 + k)
                  \    6     /                      \ 2 /      \        \2*(3 + k)//        
- ------------------------------------ + ------------------- - -----------------------------
  /       2/x*(k - pi)\\                     /       2/k*x\\          /       2/   pi*x  \\ 
  |1 + tan |----------||*(-2*pi + 2*k)   3*k*|1 + tan |---||     6*pi*|1 + tan |---------|| 
  \        \    6     //                     \        \ 2 //          \        \2*(3 + k)//
−1−tan⁡2(x(k−π)6)(2k−2π)(tan⁡2(x(k−π)6)+1)−(1−tan⁡2(πx2(k+3)))(k+3)6π(tan⁡2(πx2(k+3))+1)+1−tan⁡2(kx2)3k(tan⁡2(kx2)+1)- \frac{1 - \tan^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)}}{\left(2 k - 2 \pi\right) \left(\tan^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)} + 1\right)} - \frac{\left(1 - \tan^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)}\right) \left(k + 3\right)}{6 \pi \left(\tan^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{k x}{2} \right)}}{3 k \left(\tan^{2}{\left(\frac{k x}{2} \right)} + 1\right)} 
                1                      1              3 + k     
- ----------------------------- + ------------ - ---------------
                   /x*(k - pi)\   3*k*sec(k*x)           / pi*x\
  (-2*pi + 2*k)*sec|----------|                  6*pi*sec|-----|
                   \    3     /                          \3 + k/
−k+36πsec⁡(πxk+3)−1(2k−2π)sec⁡(x(k−π)3)+13ksec⁡(kx)- \frac{k + 3}{6 \pi \sec{\left(\frac{\pi x}{k + 3} \right)}} - \frac{1}{\left(2 k - 2 \pi\right) \sec{\left(\frac{x \left(k - \pi\right)}{3} \right)}} + \frac{1}{3 k \sec{\left(k x \right)}} 
                 2/x*(k - pi)\                      2/k*x\     /        2/   pi*x  \\        
         -1 + cot |----------|              -1 + cot |---|     |-1 + cot |---------||*(3 + k)
                  \    6     /                       \ 2 /     \         \2*(3 + k)//        
- ------------------------------------ + ------------------- - ------------------------------
  /       2/x*(k - pi)\\                     /       2/k*x\\          /       2/   pi*x  \\  
  |1 + cot |----------||*(-2*pi + 2*k)   3*k*|1 + cot |---||     6*pi*|1 + cot |---------||  
  \        \    6     //                     \        \ 2 //          \        \2*(3 + k)//
−(k+3)(cot⁡2(πx2(k+3))−1)6π(cot⁡2(πx2(k+3))+1)−cot⁡2(x(k−π)6)−1(2k−2π)(cot⁡2(x(k−π)6)+1)+cot⁡2(kx2)−13k(cot⁡2(kx2)+1)- \frac{\left(k + 3\right) \left(\cot^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)} - 1\right)}{6 \pi \left(\cot^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)} + 1\right)} - \frac{\cot^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)} - 1}{\left(2 k - 2 \pi\right) \left(\cot^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{k x}{2} \right)} - 1}{3 k \left(\cot^{2}{\left(\frac{k x}{2} \right)} + 1\right)} 
                  1                            1                  3 + k        
- ---------------------------------- + ----------------- - --------------------
                   /pi   x*(k - pi)\          /pi      \           /pi    pi*x\
  (-2*pi + 2*k)*csc|-- - ----------|   3*k*csc|-- - k*x|   6*pi*csc|-- - -----|
                   \2        3     /          \2       /           \2    3 + k/
−k+36πcsc⁡(−πxk+3+π2)−1(2k−2π)csc⁡(−x(k−π)3+π2)+13kcsc⁡(−kx+π2)- \frac{k + 3}{6 \pi \csc{\left(- \frac{\pi x}{k + 3} + \frac{\pi}{2} \right)}} - \frac{1}{\left(2 k - 2 \pi\right) \csc{\left(- \frac{x \left(k - \pi\right)}{3} + \frac{\pi}{2} \right)}} + \frac{1}{3 k \csc{\left(- k x + \frac{\pi}{2} \right)}} 
-1/((-2*pi + 2*k)*csc(pi/2 - x*(k - pi)/3)) + 1/(3*k*csc(pi/2 - k*x)) - (3 + k)/(6*pi*csc(pi/2 - pi*x/(3 + k)))
    © Sr Examen – Калькулятор