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Resolución de integrales/ (1/3)/ 3*x^2/ x^2-1/ (x*(3*x^2-1)^2)^(1/3)

Integral de (x*(3*x^2-1)^2)^(1/3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |      _______________   
 |     /             2    
 |  3 /    /   2    \     
 |  \/   x*\3*x  - 1/   dx
 |                        
/                         
0
$$\int\limits_{0}^{1} \sqrt[3]{x \left(3 x^{2} - 1\right)^{2}}\, dx$$ 
Integral((x*(3*x^2 - 1)^2)^(1/3), (x, 0, 1))
Respuesta [src] 
  1                                                                                                                                      
  /                                                                                                                                      
 |                                                                                                                                       
 |  /             2*pi*I                                              2*pi*I                                                             
 |  |             ------              _                               ------              _                                              
 |  |        7/3    3                |_  /1/3, 5/3 |    2\     3 ___    3                |_  /-2/3, 2/3 |    2\                          
 |  |     4*x   *e      *Gamma(2/3)* |   |         | 3*x |   2*\/ x *e      *Gamma(2/3)* |   |          | 3*x |                          
 |  |                               2  1 \  8/3    |     /                              2  1 \   5/3    |     /                 2        
 |  |   - ------------------------------------------------ + --------------------------------------------------     for -1 + 3*x  >= 0   
 |  |                       5*Gamma(5/3)                                        3*Gamma(5/3)                                             
 |  <                                                                                                                                  dx
 |  |                     _                                                           _                                                  
 |  |  3 ___             |_  /-2/3, 2/3 |    2  2*pi*I\      7/3  2*pi*I             |_  /1/3, 5/3 |    2  2*pi*I\                       
 |  |2*\/ x *Gamma(2/3)* |   |          | 3*x *e      |   4*x   *e      *Gamma(2/3)* |   |         | 3*x *e      |                       
 |  |                   2  1 \   5/3    |             /                             2  1 \  8/3    |             /                       
 |  |-------------------------------------------------- - --------------------------------------------------------      otherwise        
 |  |                   3*Gamma(5/3)                                            5*Gamma(5/3)                                             
 |  \                                                                                                                                    
 |                                                                                                                                       
/                                                                                                                                        
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{4 x^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {3 x^{2}} \right)}}{5 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{x} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {3 x^{2}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)} & \text{for}\: 3 x^{2} - 1 \geq 0 \\- \frac{4 x^{\frac{7}{3}} e^{2 i \pi} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {3 x^{2} e^{2 i \pi}} \right)}}{5 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{x} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {3 x^{2} e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)} & \text{otherwise} \end{cases}\, dx$$ 
=
= 
  1                                                                                                                                      
  /                                                                                                                                      
 |                                                                                                                                       
 |  /             2*pi*I                                              2*pi*I                                                             
 |  |             ------              _                               ------              _                                              
 |  |        7/3    3                |_  /1/3, 5/3 |    2\     3 ___    3                |_  /-2/3, 2/3 |    2\                          
 |  |     4*x   *e      *Gamma(2/3)* |   |         | 3*x |   2*\/ x *e      *Gamma(2/3)* |   |          | 3*x |                          
 |  |                               2  1 \  8/3    |     /                              2  1 \   5/3    |     /                 2        
 |  |   - ------------------------------------------------ + --------------------------------------------------     for -1 + 3*x  >= 0   
 |  |                       5*Gamma(5/3)                                        3*Gamma(5/3)                                             
 |  <                                                                                                                                  dx
 |  |                     _                                                           _                                                  
 |  |  3 ___             |_  /-2/3, 2/3 |    2  2*pi*I\      7/3  2*pi*I             |_  /1/3, 5/3 |    2  2*pi*I\                       
 |  |2*\/ x *Gamma(2/3)* |   |          | 3*x *e      |   4*x   *e      *Gamma(2/3)* |   |         | 3*x *e      |                       
 |  |                   2  1 \   5/3    |             /                             2  1 \  8/3    |             /                       
 |  |-------------------------------------------------- - --------------------------------------------------------      otherwise        
 |  |                   3*Gamma(5/3)                                            5*Gamma(5/3)                                             
 |  \                                                                                                                                    
 |                                                                                                                                       
/                                                                                                                                        
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{4 x^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {3 x^{2}} \right)}}{5 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{x} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {3 x^{2}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)} & \text{for}\: 3 x^{2} - 1 \geq 0 \\- \frac{4 x^{\frac{7}{3}} e^{2 i \pi} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {3 x^{2} e^{2 i \pi}} \right)}}{5 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{x} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {3 x^{2} e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)} & \text{otherwise} \end{cases}\, dx$$ 
Integral(Piecewise((-4*x^(7/3)*exp(2*pi*i/3)*gamma(2/3)*hyper((1/3, 5/3), (8/3,), 3*x^2)/(5*gamma(5/3)) + 2*x^(1/3)*exp(2*pi*i/3)*gamma(2/3)*hyper((-2/3, 2/3), (5/3,), 3*x^2)/(3*gamma(5/3)), -1 + 3*x^2 >= 0), (2*x^(1/3)*gamma(2/3)*hyper((-2/3, 2/3), (5/3,), 3*x^2*exp_polar(2*pi*i))/(3*gamma(5/3)) - 4*x^(7/3)*exp_polar(2*pi*i)*gamma(2/3)*hyper((1/3, 5/3), (8/3,), 3*x^2*exp_polar(2*pi*i))/(5*gamma(5/3)), True)), (x, 0, 1))
Respuesta numérica [src] 
0.600923929964381
0.600923929964381

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

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