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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(sqrt(2)*pi^4*K^4*(6*sqrt(2)/(pi*K)+18*sqrt(2)/(pi^3*K^3))/36)

¿Cómo vas a descomponer esta atan(sqrt(2)*pi^4*K^4*(6*sqrt(2)/(pi*K)+18*sqrt(2)/(pi^3*K^3))/36) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /             /    ___        ___\\
    |  ___   4  4 |6*\/ 2    18*\/ 2 ||
    |\/ 2 *pi *k *|------- + --------||
    |             |  pi*k       3  3 ||
    |             \           pi *k  /|
atan|---------------------------------|
    \                36               /
atan⁡(k42π4(182π3k3+62πk)36)\operatorname{atan}{\left(\frac{k^{4} \sqrt{2} \pi^{4} \left(\frac{18 \sqrt{2}}{\pi^{3} k^{3}} + \frac{6 \sqrt{2}}{\pi k}\right)}{36} \right)} 
atan((((sqrt(2)*pi^4)*k^4)*((6*sqrt(2))/((pi*k)) + (18*sqrt(2))/((pi^3*k^3))))/36)
Simplificación general [src] 
    /         3  3\
    |       pi *k |
atan|pi*k + ------|
    \         3   /
atan⁡(π3k33+πk)\operatorname{atan}{\left(\frac{\pi^{3} k^{3}}{3} + \pi k \right)} 
atan(pi*k + pi^3*k^3/3)
Descomposición de una fracción [src] 
atan(pi*k + pi^3*k^3/3)
atan⁡(π3k33+πk)\operatorname{atan}{\left(\frac{\pi^{3} k^{3}}{3} + \pi k \right)} 
    /         3  3\
    |       pi *k |
atan|pi*k + ------|
    \         3   /
Potencias [src] 
    /             /   ___       ___ \\
    |  ___   4  4 | \/ 2      \/ 2  ||
atan|\/ 2 *pi *k *|-------- + ------||
    |             |    3  3   6*pi*k||
    \             \2*pi *k          //
atan⁡(2π4k4(26πk+22π3k3))\operatorname{atan}{\left(\sqrt{2} \pi^{4} k^{4} \left(\frac{\sqrt{2}}{6 \pi k} + \frac{\sqrt{2}}{2 \pi^{3} k^{3}}\right) \right)} 
    /             /    ___        ___\\
    |  ___   4  4 |6*\/ 2    18*\/ 2 ||
    |\/ 2 *pi *k *|------- + --------||
    |             |  pi*k       3  3 ||
    |             \           pi *k  /|
atan|---------------------------------|
    \                36               /
atan⁡(2π4k4(62πk+182π3k3)36)\operatorname{atan}{\left(\frac{\sqrt{2} \pi^{4} k^{4} \left(\frac{6 \sqrt{2}}{\pi k} + \frac{18 \sqrt{2}}{\pi^{3} k^{3}}\right)}{36} \right)} 
atan(sqrt(2)*pi^4*k^4*(6*sqrt(2)/(pi*k) + 18*sqrt(2)/(pi^3*k^3))/36)
Respuesta numérica [src] 
atan((((sqrt(2)*pi^4)*k^4)*((6*sqrt(2))/((pi*k)) + (18*sqrt(2))/((pi^3*k^3))))/36)
atan((((sqrt(2)*pi^4)*k^4)*((6*sqrt(2))/((pi*k)) + (18*sqrt(2))/((pi^3*k^3))))/36)
Parte trigonométrica [src] 
    /             /    ___        ___\\
    |  ___   4  4 |6*\/ 2    18*\/ 2 ||
    |\/ 2 *pi *k *|------- + --------||
    |             |  pi*k       3  3 ||
    |             \           pi *k  /|
atan|---------------------------------|
    \                36               /
atan⁡(2π4k4(62πk+182π3k3)36)\operatorname{atan}{\left(\frac{\sqrt{2} \pi^{4} k^{4} \left(\frac{6 \sqrt{2}}{\pi k} + \frac{18 \sqrt{2}}{\pi^{3} k^{3}}\right)}{36} \right)} 
atan(sqrt(2)*pi^4*k^4*(6*sqrt(2)/(pi*k) + 18*sqrt(2)/(pi^3*k^3))/36)
Unión de expresiones racionales [src] 
    /     /      2  2\\
    |pi*k*\3 + pi *k /|
atan|-----------------|
    \        3        /
atan⁡(πk(π2k2+3)3)\operatorname{atan}{\left(\frac{\pi k \left(\pi^{2} k^{2} + 3\right)}{3} \right)} 
atan(pi*k*(3 + pi^2*k^2)/3)
Abrimos la expresión [src] 
    /             /    ___        ___\\
    |  ___   4  4 |6*\/ 2    18*\/ 2 ||
    |\/ 2 *pi *k *|------- + --------||
    |             |  pi*k       3  3 ||
    |             \           pi *k  /|
atan|---------------------------------|
    \                36               /
atan⁡(2π4k4(62πk+182π3k3)36)\operatorname{atan}{\left(\frac{\sqrt{2} \pi^{4} k^{4} \left(\frac{6 \sqrt{2}}{\pi k} + \frac{18 \sqrt{2}}{\pi^{3} k^{3}}\right)}{36} \right)} 
atan(sqrt(2)*pi^4*k^4*(6*sqrt(2)/(pi*k) + 18*sqrt(2)/(pi^3*k^3))/36)
Combinatoria [src] 
    /         3  3\
    |       pi *k |
atan|pi*k + ------|
    \         3   /
atan⁡(π3k33+πk)\operatorname{atan}{\left(\frac{\pi^{3} k^{3}}{3} + \pi k \right)} 
atan(pi*k + pi^3*k^3/3)
Denominador común [src] 
    /         3  3\
    |       pi *k |
atan|pi*k + ------|
    \         3   /
atan⁡(π3k33+πk)\operatorname{atan}{\left(\frac{\pi^{3} k^{3}}{3} + \pi k \right)} 
atan(pi*k + pi^3*k^3/3)
Denominador racional [src] 
    /     /      2  2\\
    |pi*k*\3 + pi *k /|
atan|-----------------|
    \        3        /
atan⁡(πk(π2k2+3)3)\operatorname{atan}{\left(\frac{\pi k \left(\pi^{2} k^{2} + 3\right)}{3} \right)} 
atan(pi*k*(3 + pi^2*k^2)/3)
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