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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  5                 
  /                 
 |                  
 |       ________   
 |      /      3    
 |  x*\/  1 - x   dx
 |                  
/                   
4
$$\int\limits_{4}^{5} x \sqrt{1 - x^{3}}\, dx$$ 
Integral(x*sqrt(1 - x^3), (x, 4, 5))
Respuesta (Indefinida) [src] 
  /                                       _                          
 |                         2             |_  /-1/2, 2/3 |  3  2*pi*I\
 |      ________          x *Gamma(2/3)* |   |          | x *e      |
 |     /      3                         2  1 \   5/3    |           /
 | x*\/  1 - x   dx = C + -------------------------------------------
 |                                        3*Gamma(5/3)               
/
$$\int x \sqrt{1 - x^{3}}\, dx = C + \frac{x^{2} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3} e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$$
Simplificar
Respuesta [src] 
                  _                                             _                           
                 |_  /-1/2, 2/3 |     2*pi*I\                  |_  /-1/2, 2/3 |      2*pi*I\
  16*Gamma(2/3)* |   |          | 64*e      |   25*Gamma(2/3)* |   |          | 125*e      |
                2  1 \   5/3    |           /                 2  1 \   5/3    |            /
- ------------------------------------------- + --------------------------------------------
                  3*Gamma(5/3)                                  3*Gamma(5/3)
$$- \frac{16 \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {64 e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)} + \frac{25 \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {125 e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$$ 
=
= 
                  _                                             _                           
                 |_  /-1/2, 2/3 |     2*pi*I\                  |_  /-1/2, 2/3 |      2*pi*I\
  16*Gamma(2/3)* |   |          | 64*e      |   25*Gamma(2/3)* |   |          | 125*e      |
                2  1 \   5/3    |           /                 2  1 \   5/3    |            /
- ------------------------------------------- + --------------------------------------------
                  3*Gamma(5/3)                                  3*Gamma(5/3)
$$- \frac{16 \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {64 e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)} + \frac{25 \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {125 e^{2 i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$$ 
-16*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), 64*exp_polar(2*pi*i))/(3*gamma(5/3)) + 25*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), 125*exp_polar(2*pi*i))/(3*gamma(5/3))
Respuesta numérica [src] 
(0.0 + 43.0514020507714j)
(0.0 + 43.0514020507714j)

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

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