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Resolución de integrales/ x-x^3/ sqrt(16x-x^3-y^3)

Integral de sqrt(16x-x^3-y^3) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                       
  /                       
 |                        
 |     ________________   
 |    /         3    3    
 |  \/  16*x - x  - y   dy
 |                        
/                         
0
$$\int\limits_{0}^{2} \sqrt{- y^{3} + \left(- x^{3} + 16 x\right)}\, dy$$ 
Integral(sqrt(16*x - x^3 - y^3), (y, 0, 2))
Respuesta (Indefinida) [src] 
                                                                                                                    
                                     _________________________              _  /          |         3  2*pi*I      \
  /                                 /           /   3       \              |_  |-1/2, 1/3 |        y *e            |
 |                              y*\/  polar_lift\- x  + 16*x/ *Gamma(1/3)* |   |          | -----------------------|
 |    ________________                                                    2  1 |   4/3    |           /   3       \|
 |   /         3    3                                                          \          | polar_lift\- x  + 16*x//
 | \/  16*x - x  - y   dy = C + ------------------------------------------------------------------------------------
 |                                                                  3*Gamma(4/3)                                    
/
$$\int \sqrt{- y^{3} + \left(- x^{3} + 16 x\right)}\, dy = C + \frac{y \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{y^{3} e^{2 i \pi}}{\operatorname{polar\_lift}{\left(- x^{3} + 16 x \right)}}} \right)} \sqrt{\operatorname{polar\_lift}{\left(- x^{3} + 16 x \right)}}}{3 \Gamma\left(\frac{4}{3}\right)}$$
Simplificar
Respuesta [src] 
                                                                                    
     _________________________              _  /          |           2*pi*I       \
    /           /   3       \              |_  |-1/2, 1/3 |        8*e             |
2*\/  polar_lift\- x  + 16*x/ *Gamma(1/3)* |   |          | -----------------------|
                                          2  1 |   4/3    |           /   3       \|
                                               \          | polar_lift\- x  + 16*x//
------------------------------------------------------------------------------------
                                    3*Gamma(4/3)
$$\frac{2 \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{8 e^{2 i \pi}}{\operatorname{polar\_lift}{\left(- x^{3} + 16 x \right)}}} \right)} \sqrt{\operatorname{polar\_lift}{\left(- x^{3} + 16 x \right)}}}{3 \Gamma\left(\frac{4}{3}\right)}$$ 
=
= 
                                                                                    
     _________________________              _  /          |           2*pi*I       \
    /           /   3       \              |_  |-1/2, 1/3 |        8*e             |
2*\/  polar_lift\- x  + 16*x/ *Gamma(1/3)* |   |          | -----------------------|
                                          2  1 |   4/3    |           /   3       \|
                                               \          | polar_lift\- x  + 16*x//
------------------------------------------------------------------------------------
                                    3*Gamma(4/3)
$$\frac{2 \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{8 e^{2 i \pi}}{\operatorname{polar\_lift}{\left(- x^{3} + 16 x \right)}}} \right)} \sqrt{\operatorname{polar\_lift}{\left(- x^{3} + 16 x \right)}}}{3 \Gamma\left(\frac{4}{3}\right)}$$ 
2*sqrt(polar_lift(-x^3 + 16*x))*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), 8*exp_polar(2*pi*i)/polar_lift(-x^3 + 16*x))/(3*gamma(4/3))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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