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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(sin(x)/(sqrt(3)*(1+cos(x))))/(2*sqrt(3))

¿Cómo vas a descomponer esta atan(sin(x)/(sqrt(3)*(1+cos(x))))/(2*sqrt(3)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /      sin(x)      \
atan|------------------|
    |  ___             |
    \\/ 3 *(1 + cos(x))/
------------------------
            ___         
        2*\/ 3
$$\frac{\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{\sqrt{3} \left(\cos{\left(x \right)} + 1\right)} \right)}}{2 \sqrt{3}}$$ 
atan(sin(x)/((sqrt(3)*(1 + cos(x)))))/((2*sqrt(3)))
Descomposición de una fracción [src] 
sqrt(3)*atan(sin(x)/(sqrt(3) + sqrt(3)*cos(x)))/6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{\sqrt{3} \cos{\left(x \right)} + \sqrt{3}} \right)}}{6}$$ 
  ___     /       sin(x)       \
\/ 3 *atan|--------------------|
          |  ___     ___       |
          \\/ 3  + \/ 3 *cos(x)/
--------------------------------
               6
Simplificación general [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sqrt(3)*sin(x)/(3*(1 + cos(x))))/6
Respuesta numérica [src] 
0.288675134594813*atan(sin(x)/((sqrt(3)*(1 + cos(x)))))
0.288675134594813*atan(sin(x)/((sqrt(3)*(1 + cos(x)))))
Combinatoria [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sqrt(3)*sin(x)/(3*(1 + cos(x))))/6
Denominador común [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sqrt(3)*sin(x)/(3*(1 + cos(x))))/6
Denominador racional [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sqrt(3)*sin(x)/(3*(1 + cos(x))))/6
Potencias [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
           /    ___ /   -I*x    I*x\\ 
   ___     |I*\/ 3 *\- e     + e   /| 
-\/ 3 *atan|------------------------| 
           |    /     I*x    -I*x\  | 
           |    |    e      e    |  | 
           |  6*|1 + ---- + -----|  | 
           \    \     2       2  /  / 
--------------------------------------
                  6
$$- \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} i \left(e^{i x} - e^{- i x}\right)}{6 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)} \right)}}{6}$$ 
-sqrt(3)*atan(i*sqrt(3)*(-exp(-i*x) + exp(i*x))/(6*(1 + exp(i*x)/2 + exp(-i*x)/2)))/6
Unión de expresiones racionales [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sqrt(3)*sin(x)/(3*(1 + cos(x))))/6
Compilar la expresión [src] 
  ___     /      sin(x)      \
\/ 3 *atan|------------------|
          |  ___             |
          \\/ 3 *(1 + cos(x))/
------------------------------
              6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{\sqrt{3} \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sin(x)/((sqrt(3)*(1 + cos(x)))))/6
Abrimos la expresión [src] 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
  ___                           
\/ 3      /       sin(x)       \
-----*atan|--------------------|
  6       |  ___     ___       |
          \\/ 3  + \/ 3 *cos(x)/
$$\frac{\sqrt{3}}{6} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{\sqrt{3} \cos{\left(x \right)} + \sqrt{3}} \right)}$$ 
(sqrt(3)/6)*atan(sin(x)/(sqrt(3) + sqrt(3)*cos(x)))
Parte trigonométrica [src] 
          /            ___           \
  ___     |          \/ 3            |
\/ 3 *atan|--------------------------|
          |  /         1     \       |
          |3*|1 + -----------|*csc(x)|
          |  |       /pi    \|       |
          |  |    csc|-- - x||       |
          \  \       \2     //       /
--------------------------------------
                  6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3 \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} \right)}}{6}$$ 
          /      ___          \
  ___     |    \/ 3 *sin(x)   |
\/ 3 *atan|-------------------|
          |  /       /    pi\\|
          |3*|1 + sin|x + --|||
          \  \       \    2 ///
-------------------------------
               6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} \right)}}{6}$$ 
          /            ___           \
  ___     |          \/ 3            |
\/ 3 *atan|--------------------------|
          |  /      1   \    /    pi\|
          |3*|1 + ------|*sec|x - --||
          \  \    sec(x)/    \    2 //
--------------------------------------
                  6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3 \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{6}$$ 
          /          ___        \
  ___     |        \/ 3         |
\/ 3 *atan|---------------------|
          |  /      1   \       |
          |3*|1 + ------|*csc(x)|
          \  \    sec(x)/       /
---------------------------------
                6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3 \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}} \right)}}{6}$$ 
          /              ___    /x\          \
          |          2*\/ 3 *cot|-|          |
  ___     |                     \2/          |
\/ 3 *atan|----------------------------------|
          |                /            2/x\\|
          |                |    -1 + cot |-|||
          |  /       2/x\\ |             \2/||
          |3*|1 + cot |-||*|1 + ------------||
          |  \        \2// |           2/x\ ||
          |                |    1 + cot |-| ||
          \                \            \2/ //
----------------------------------------------
                      6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \cot{\left(\frac{x}{2} \right)}}{3 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)}}{6}$$ 
          /  ___    /    pi\\
          |\/ 3 *cos|x - --||
  ___     |         \    2 /|
\/ 3 *atan|-----------------|
          \  3*(1 + cos(x)) /
-----------------------------
              6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \cos{\left(x - \frac{\pi}{2} \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
          /   ___        \
  ___     | \/ 3 *sin(x) |
\/ 3 *atan|--------------|
          \3*(1 + cos(x))/
--------------------------
            6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \sin{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
          /              ___    /x\         \
          |          2*\/ 3 *tan|-|         |
  ___     |                     \2/         |
\/ 3 *atan|---------------------------------|
          |                /           2/x\\|
          |                |    1 - tan |-|||
          |  /       2/x\\ |            \2/||
          |3*|1 + tan |-||*|1 + -----------||
          |  \        \2// |           2/x\||
          |                |    1 + tan |-|||
          \                \            \2///
---------------------------------------------
                      6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} \right)}}{6}$$ 
  ___     /      sin(x)      \
\/ 3 *atan|------------------|
          |  ___             |
          \\/ 3 *(1 + cos(x))/
------------------------------
              6
$$\frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{\sqrt{3} \left(\cos{\left(x \right)} + 1\right)} \right)}}{6}$$ 
sqrt(3)*atan(sin(x)/((sqrt(3)*(1 + cos(x)))))/6
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