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Resolución de integrales/ 5-x^3/ 1/sqrt/ 1/sqrt(5-x^3)

Integral de 1/sqrt(5-x^3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2               
  /               
 |                
 |       1        
 |  ----------- dx
 |     ________   
 |    /      3    
 |  \/  5 - x     
 |                
/                 
-1
$$\int\limits_{-1}^{2} \frac{1}{\sqrt{5 - x^{3}}}\, dx$$ 
Integral(1/(sqrt(5 - x^3)), (x, -1, 2))
Respuesta (Indefinida) [src] 
                                                                       
                                             _  /         |  3  2*pi*I\
  /                         ___             |_  |1/3, 1/2 | x *e      |
 |                      x*\/ 5 *Gamma(1/3)* |   |         | ----------|
 |      1                                  2  1 \  4/3    |     5     /
 | ----------- dx = C + -----------------------------------------------
 |    ________                           15*Gamma(4/3)                 
 |   /      3                                                          
 | \/  5 - x                                                           
 |                                                                     
/
$$\int \frac{1}{\sqrt{5 - x^{3}}}\, dx = C + \frac{\sqrt{5} x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{5}} \right)}}{15 \Gamma\left(\frac{4}{3}\right)}$$
Simplificar
Respuesta [src] 
                                                                                        
                   _                                           _  /         |    2*pi*I\
  ___             |_  /1/3, 1/2 |     \       ___             |_  |1/3, 1/2 | 8*e      |
\/ 5 *Gamma(1/3)* |   |         | -1/5|   2*\/ 5 *Gamma(1/3)* |   |         | ---------|
                 2  1 \  4/3    |     /                      2  1 \  4/3    |     5    /
--------------------------------------- + ----------------------------------------------
             15*Gamma(4/3)                                15*Gamma(4/3)
$$\frac{\sqrt{5} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {- \frac{1}{5}} \right)}}{15 \Gamma\left(\frac{4}{3}\right)} + \frac{2 \sqrt{5} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {\frac{8 e^{2 i \pi}}{5}} \right)}}{15 \Gamma\left(\frac{4}{3}\right)}$$ 
=
= 
                                                                                        
                   _                                           _  /         |    2*pi*I\
  ___             |_  /1/3, 1/2 |     \       ___             |_  |1/3, 1/2 | 8*e      |
\/ 5 *Gamma(1/3)* |   |         | -1/5|   2*\/ 5 *Gamma(1/3)* |   |         | ---------|
                 2  1 \  4/3    |     /                      2  1 \  4/3    |     5    /
--------------------------------------- + ----------------------------------------------
             15*Gamma(4/3)                                15*Gamma(4/3)
$$\frac{\sqrt{5} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {- \frac{1}{5}} \right)}}{15 \Gamma\left(\frac{4}{3}\right)} + \frac{2 \sqrt{5} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle| {\frac{8 e^{2 i \pi}}{5}} \right)}}{15 \Gamma\left(\frac{4}{3}\right)}$$ 
sqrt(5)*gamma(1/3)*hyper((1/3, 1/2), (4/3,), -1/5)/(15*gamma(4/3)) + 2*sqrt(5)*gamma(1/3)*hyper((1/3, 1/2), (4/3,), 8*exp_polar(2*pi*i)/5)/(15*gamma(4/3))
Respuesta numérica [src] 
(1.49984350839026 - 0.315835260228914j)
(1.49984350839026 - 0.315835260228914j)

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

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