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Simplificación de expresiones/ Descomposición en fracciones simples/ cot(x^2/2+pi/3)/(-1+cot(x^2/2+pi/3)^2)

¿Cómo vas a descomponer esta cot(x^2/2+pi/3)/(-1+cot(x^2/2+pi/3)^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
      / 2     \   
      |x    pi|   
   cot|-- + --|   
      \2    3 /   
------------------
         / 2     \
        2|x    pi|
-1 + cot |-- + --|
         \2    3 /
cot⁡(x22+π3)cot⁡2(x22+π3)−1\frac{\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)}}{\cot^{2}{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)} - 1} 
cot(x^2/2 + pi/3)/(-1 + cot(x^2/2 + pi/3)^2)
Descomposición de una fracción [src] 
1/(2*(1 + cot(x^2/2 + pi/3))) + 1/(2*(-1 + cot(x^2/2 + pi/3)))
12(cot⁡(x22+π3)+1)+12(cot⁡(x22+π3)−1)\frac{1}{2 \left(\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)} + 1\right)} + \frac{1}{2 \left(\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)} - 1\right)} 
         1                       1          
-------------------- + ---------------------
  /       / 2     \\     /        / 2     \\
  |       |x    pi||     |        |x    pi||
2*|1 + cot|-- + --||   2*|-1 + cot|-- + --||
  \       \2    3 //     \        \2    3 //
Simplificación general [src] 
      / 2     \   
      |x    pi|   
   cot|-- + --|   
      \2    3 /   
------------------
           1      
-1 + -------------
         / 2     \
        2|x    pi|
     tan |-- + --|
         \2    3 /
cot⁡(x22+π3)−1+1tan⁡2(x22+π3)\frac{\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)}}{-1 + \frac{1}{\tan^{2}{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)}}} 
cot(x^2/2 + pi/3)/(-1 + tan(x^2/2 + pi/3)^(-2))
Denominador racional [src] 
     / 2   2*pi\  
     |x  + ----|  
     |      3  |  
  cot|---------|  
     \    2    /  
------------------
         / 2     \
        2|x    pi|
-1 + cot |-- + --|
         \2    3 /
cot⁡(x2+2π32)cot⁡2(x22+π3)−1\frac{\cot{\left(\frac{x^{2} + \frac{2 \pi}{3}}{2} \right)}}{\cot^{2}{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)} - 1} 
cot((x^2 + 2*pi/3)/2)/(-1 + cot(x^2/2 + pi/3)^2)
Respuesta numérica [src] 
cot(x^2/2 + pi/3)/(-1.0 + cot(x^2/2 + pi/3)^2)
cot(x^2/2 + pi/3)/(-1.0 + cot(x^2/2 + pi/3)^2)
Unión de expresiones racionales [src] 
      /          2\   
      |2*pi + 3*x |   
   cot|-----------|   
      \     6     /   
----------------------
         /          2\
        2|2*pi + 3*x |
-1 + cot |-----------|
         \     6     /
cot⁡(3x2+2π6)cot⁡2(3x2+2π6)−1\frac{\cot{\left(\frac{3 x^{2} + 2 \pi}{6} \right)}}{\cot^{2}{\left(\frac{3 x^{2} + 2 \pi}{6} \right)} - 1} 
cot((2*pi + 3*x^2)/6)/(-1 + cot((2*pi + 3*x^2)/6)^2)
Abrimos la expresión [src] 
                                                                                                                                                                                                                                                                 / 2\                                                                              
                                                                                                                                                                                                                                                          ___    |x |                                                                              
                                                                                                                                                                                                                                                        \/ 3 *cot|--|                                                                              
                                                                                    1                                                                                                                                                                            \2 /                                                                              
- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                            / 2\                                / 2\                                      / 2\              /                                                                          / 2\                                / 2\                                      / 2\         \
                                                                           3|x |                                |x |                              ___    2|x |              |                                                                         3|x |                                |x |                              ___    2|x |         |
       / 2\     ___                   ___                               cot |--|                             cot|--|                          5*\/ 3 *cot |--|              |     / 2\     ___                   ___                               cot |--|                             cot|--|                          5*\/ 3 *cot |--|         |
       |x |   \/ 3                  \/ 3                                    \2 /                                \2 /                                      \2 /              |     |x |   \/ 3                  \/ 3                                    \2 /                                \2 /                                      \2 /         |
  - cot|--| - ----- + ---------------------------------- + ---------------------------------- + ---------------------------------- - ----------------------------------   3*|- cot|--| - ----- + ---------------------------------- + ---------------------------------- + ---------------------------------- - ----------------------------------|
       \2 /     3       /                          / 2\\     /                          / 2\\     /                          / 2\\     /                          / 2\\     |     \2 /     3       /                          / 2\\     /                          / 2\\     /                          / 2\\     /                          / 2\\|
                        |                   ___    |x ||     |                   ___    |x ||     |                   ___    |x ||     |                   ___    |x ||     |                      |                   ___    |x ||     |                   ___    |x ||     |                   ___    |x ||     |                   ___    |x |||
                        |        / 2\   2*\/ 3 *cot|--||     |        / 2\   2*\/ 3 *cot|--||     |        / 2\   2*\/ 3 *cot|--||     |        / 2\   2*\/ 3 *cot|--||     |                      |        / 2\   2*\/ 3 *cot|--||     |        / 2\   2*\/ 3 *cot|--||     |        / 2\   2*\/ 3 *cot|--||     |        / 2\   2*\/ 3 *cot|--|||
                        |1      2|x |              \2 /|     |1      2|x |              \2 /|     |1      2|x |              \2 /|     |1      2|x |              \2 /|     |                      |1      2|x |              \2 /|     |1      2|x |              \2 /|     |1      2|x |              \2 /|     |1      2|x |              \2 /||
                      3*|- + cot |--| + ---------------|   3*|- + cot |--| + ---------------|   3*|- + cot |--| + ---------------|   9*|- + cot |--| + ---------------|     |                    3*|- + cot |--| + ---------------|   3*|- + cot |--| + ---------------|   3*|- + cot |--| + ---------------|   9*|- + cot |--| + ---------------||
                        \3       \2 /          3       /     \3       \2 /          3       /     \3       \2 /          3       /     \3       \2 /          3       /     \                      \3       \2 /          3       /     \3       \2 /          3       /     \3       \2 /          3       /     \3       \2 /          3       //
3cot⁡(x22)3(−cot⁡(x22)−33+cot⁡3(x22)3(cot⁡2(x22)+23cot⁡(x22)3+13)−53cot⁡2(x22)9(cot⁡2(x22)+23cot⁡(x22)3+13)+cot⁡(x22)3(cot⁡2(x22)+23cot⁡(x22)3+13)+33(cot⁡2(x22)+23cot⁡(x22)3+13))−1−cot⁡(x22)−33+cot⁡3(x22)3(cot⁡2(x22)+23cot⁡(x22)3+13)−53cot⁡2(x22)9(cot⁡2(x22)+23cot⁡(x22)3+13)+cot⁡(x22)3(cot⁡2(x22)+23cot⁡(x22)3+13)+33(cot⁡2(x22)+23cot⁡(x22)3+13)\frac{\sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3 \left(- \cot{\left(\frac{x^{2}}{2} \right)} - \frac{\sqrt{3}}{3} + \frac{\cot^{3}{\left(\frac{x^{2}}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)} - \frac{5 \sqrt{3} \cot^{2}{\left(\frac{x^{2}}{2} \right)}}{9 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)} + \frac{\cot{\left(\frac{x^{2}}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)} + \frac{\sqrt{3}}{3 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)}\right)} - \frac{1}{- \cot{\left(\frac{x^{2}}{2} \right)} - \frac{\sqrt{3}}{3} + \frac{\cot^{3}{\left(\frac{x^{2}}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)} - \frac{5 \sqrt{3} \cot^{2}{\left(\frac{x^{2}}{2} \right)}}{9 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)} + \frac{\cot{\left(\frac{x^{2}}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)} + \frac{\sqrt{3}}{3 \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + \frac{2 \sqrt{3} \cot{\left(\frac{x^{2}}{2} \right)}}{3} + \frac{1}{3}\right)}} 
-1/(-cot(x^2/2) - sqrt(3)/3 + sqrt(3)/(3*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3)) + cot(x^2/2)^3/(3*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3)) + cot(x^2/2)/(3*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3)) - 5*sqrt(3)*cot(x^2/2)^2/(9*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3))) + sqrt(3)*cot(x^2/2)/(3*(-cot(x^2/2) - sqrt(3)/3 + sqrt(3)/(3*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3)) + cot(x^2/2)^3/(3*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3)) + cot(x^2/2)/(3*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3)) - 5*sqrt(3)*cot(x^2/2)^2/(9*(1/3 + cot(x^2/2)^2 + 2*sqrt(3)*cot(x^2/2)/3))))
Combinatoria [src] 
                / 2     \             
                |x    pi|             
             cot|-- + --|             
                \2    3 /             
--------------------------------------
/       / 2     \\ /        / 2     \\
|       |x    pi|| |        |x    pi||
|1 + cot|-- + --||*|-1 + cot|-- + --||
\       \2    3 // \        \2    3 //
cot⁡(x22+π3)(cot⁡(x22+π3)−1)(cot⁡(x22+π3)+1)\frac{\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)}}{\left(\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)} - 1\right) \left(\cot{\left(\frac{x^{2}}{2} + \frac{\pi}{3} \right)} + 1\right)} 
cot(x^2/2 + pi/3)/((1 + cot(x^2/2 + pi/3))*(-1 + cot(x^2/2 + pi/3)))
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