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Resolución de integrales/ 3*x^3/ sqrt(1+3*x^3)

Integral de sqrt(1+3*x^3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |     __________   
 |    /        3    
 |  \/  1 + 3*x   dx
 |                  
/                   
-1
$$\int\limits_{-1}^{1} \sqrt{3 x^{3} + 1}\, dx$$ 
Integral(sqrt(1 + 3*x^3), (x, -1, 1))
Respuesta (Indefinida) [src] 
  /                                      _                          
 |                                      |_  /-1/2, 1/3 |    3  pi*I\
 |    __________          x*Gamma(1/3)* |   |          | 3*x *e    |
 |   /        3                        2  1 \   4/3    |           /
 | \/  1 + 3*x   dx = C + ------------------------------------------
 |                                       3*Gamma(4/3)               
/
$$\int \sqrt{3 x^{3} + 1}\, dx = C + \frac{x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {3 x^{3} e^{i \pi}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
             _                                        _                       
            |_  /-1/2, 1/3 |     pi*I\               |_  /-1/2, 1/3 |    pi*I\
Gamma(1/3)* |   |          | -3*e    |   Gamma(1/3)* |   |          | 3*e    |
           2  1 \   4/3    |         /              2  1 \   4/3    |        /
-------------------------------------- + -------------------------------------
             3*Gamma(4/3)                             3*Gamma(4/3)
$$\frac{\Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {- 3 e^{i \pi}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{\Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {3 e^{i \pi}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$ 
=
= 
             _                                        _                       
            |_  /-1/2, 1/3 |     pi*I\               |_  /-1/2, 1/3 |    pi*I\
Gamma(1/3)* |   |          | -3*e    |   Gamma(1/3)* |   |          | 3*e    |
           2  1 \   4/3    |         /              2  1 \   4/3    |        /
-------------------------------------- + -------------------------------------
             3*Gamma(4/3)                             3*Gamma(4/3)
$$\frac{\Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {- 3 e^{i \pi}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{\Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {3 e^{i \pi}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$ 
gamma(1/3)*hyper((-1/2, 1/3), (4/3,), -3*exp_polar(pi*i))/(3*gamma(4/3)) + gamma(1/3)*hyper((-1/2, 1/3), (4/3,), 3*exp_polar(pi*i))/(3*gamma(4/3))
Respuesta numérica [src] 
(1.87213289355168 + 0.267311446397984j)
(1.87213289355168 + 0.267311446397984j)

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

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