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Resolución de integrales/ 2*t^2/ t^2+1/ sqrt(9*x^4-2*t^2+1)

Integral de sqrt(9*x^4-2*t^2+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |     _________________   
 |    /    4      2        
 |  \/  9*x  - 2*t  + 1  dx
 |                         
/                          
-1
$$\int\limits_{-1}^{1} \sqrt{\left(- 2 t^{2} + 9 x^{4}\right) + 1}\, dx$$ 
Integral(sqrt(9*x^4 - 2*t^2 + 1), (x, -1, 1))
Respuesta (Indefinida) [src] 
                                                                                                               
                                      ______________________              _  /          |         4  pi*I     \
  /                                  /           /       2\              |_  |-1/2, 1/4 |      9*x *e         |
 |                               x*\/  polar_lift\1 - 2*t / *Gamma(1/4)* |   |          | --------------------|
 |    _________________                                                 2  1 |   5/4    |           /       2\|
 |   /    4      2                                                           \          | polar_lift\1 - 2*t //
 | \/  9*x  - 2*t  + 1  dx = C + ------------------------------------------------------------------------------
 |                                                                4*Gamma(5/4)                                 
/
$$\int \sqrt{\left(- 2 t^{2} + 9 x^{4}\right) + 1}\, dx = C + \frac{x \Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle| {\frac{9 x^{4} e^{i \pi}}{\operatorname{polar\_lift}{\left(1 - 2 t^{2} \right)}}} \right)} \sqrt{\operatorname{polar\_lift}{\left(1 - 2 t^{2} \right)}}}{4 \Gamma\left(\frac{5}{4}\right)}$$
Simplificar
Respuesta [src] 
                                                                            
   ______________________              _  /          |          pi*I       \
  /           /       2\              |_  |-1/2, 1/4 |       9*e           |
\/  polar_lift\1 - 2*t / *Gamma(1/4)* |   |          | --------------------|
                                     2  1 |   5/4    |           /       2\|
                                          \          | polar_lift\1 - 2*t //
----------------------------------------------------------------------------
                                2*Gamma(5/4)
$$\frac{\Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle| {\frac{9 e^{i \pi}}{\operatorname{polar\_lift}{\left(1 - 2 t^{2} \right)}}} \right)} \sqrt{\operatorname{polar\_lift}{\left(1 - 2 t^{2} \right)}}}{2 \Gamma\left(\frac{5}{4}\right)}$$ 
=
= 
                                                                            
   ______________________              _  /          |          pi*I       \
  /           /       2\              |_  |-1/2, 1/4 |       9*e           |
\/  polar_lift\1 - 2*t / *Gamma(1/4)* |   |          | --------------------|
                                     2  1 |   5/4    |           /       2\|
                                          \          | polar_lift\1 - 2*t //
----------------------------------------------------------------------------
                                2*Gamma(5/4)
$$\frac{\Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle| {\frac{9 e^{i \pi}}{\operatorname{polar\_lift}{\left(1 - 2 t^{2} \right)}}} \right)} \sqrt{\operatorname{polar\_lift}{\left(1 - 2 t^{2} \right)}}}{2 \Gamma\left(\frac{5}{4}\right)}$$ 
sqrt(polar_lift(1 - 2*t^2))*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), 9*exp_polar(pi*i)/polar_lift(1 - 2*t^2))/(2*gamma(5/4))

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

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