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Expresiones semejantes
(x^5+3x-1)/(x^3-1)^(2/5)
(x^5+3x+1)/(x^3+1)^(2/5)
(x^5-3x+1)/(x^3-1)^(2/5)

Resolución de integrales/ x^3-1/ (x^5+3x+1)/(x^3-1)^(2/5)

Integral de (x^5+3x+1)/(x^3-1)^(2/5) dx

Solución

Ha introducido [src]

  3                
  /                
 |                 
 |   5             
 |  x  + 3*x + 1   
 |  ------------ dx
 |          2/5    
 |  / 3    \       
 |  \x  - 1/       
 |                 
/                  
1

$$\int\limits_{1}^{3} \frac{\left(x^{5} + 3 x\right) + 1}{\left(x^{3} - 1\right)^{\frac{2}{5}}}\, dx$$

Integral((x^5 + 3*x + 1)/(x^3 - 1)^(2/5), (x, 1, 3))

Solución detallada

Vuelva a escribir el integrando:

$\frac{\left(x^{5} + 3 x\right) + 1}{\left(x^{3} - 1\right)^{\frac{2}{5}}} = \frac{x^{5}}{\left(x^{3} - 1\right)^{\frac{2}{5}}} + \frac{3 x}{\left(x^{3} - 1\right)^{\frac{2}{5}}} + \frac{1}{\left(x^{3} - 1\right)^{\frac{2}{5}}}$
Integramos término a término:
1. No puedo encontrar los pasos en la búsqueda de esta integral.
  
  Pero la integral
  
  $\begin{cases} \frac{5 x^{3} \left(x^{3} - 1\right)^{\frac{3}{5}}}{24} + \frac{25 \left(x^{3} - 1\right)^{\frac{3}{5}}}{72} & \text{for}\: \left|{x^{3}}\right| > 1 \\- \frac{5 x^{3} \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{24} - \frac{25 \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{72} & \text{otherwese} \end{cases}$
1. La integral del producto de una función por una constante es la constante por la integral de esta función:
  
  $\int \frac{3 x}{\left(x^{3} - 1\right)^{\frac{2}{5}}}\, dx = 3 \int \frac{x}{\left(x^{3} - 1\right)^{\frac{2}{5}}}\, dx$
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $\frac{x^{2} e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$
  Por lo tanto, el resultado es: $\frac{x^{2} e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)}$
1. No puedo encontrar los pasos en la búsqueda de esta integral.
  
  Pero la integral
  
  $\frac{x e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$
El resultado es: $\frac{x^{2} e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} + \frac{x e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \begin{cases} \frac{5 x^{3} \left(x^{3} - 1\right)^{\frac{3}{5}}}{24} + \frac{25 \left(x^{3} - 1\right)^{\frac{3}{5}}}{72} & \text{for}\: \left|{x^{3}}\right| > 1 \\- \frac{5 x^{3} \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{24} - \frac{25 \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{72} & \text{otherwese} \end{cases}$
Ahora simplificar:

$\begin{cases} \frac{\sqrt{3} \left(\frac{5 x^{3} \left(x^{3} - 1\right)^{\frac{3}{5}}}{3} - 12 \left(-1\right)^{\frac{3}{5}} x^{2} {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)} - 8 \left(-1\right)^{\frac{3}{5}} x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)} + \frac{25 \left(x^{3} - 1\right)^{\frac{3}{5}}}{9}\right) \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}{16 \pi} & \text{for}\: \left|{x^{3}}\right| > 1 \\\frac{\left(-1\right)^{\frac{3}{5}} \sqrt{3} \left(\frac{5 x^{3} \left(1 - x^{3}\right)^{\frac{3}{5}}}{3} - 12 x^{2} {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)} - 8 x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)} + \frac{25 \left(1 - x^{3}\right)^{\frac{3}{5}}}{9}\right) \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}{16 \pi} & \text{otherwese} \end{cases}$
Añadimos la constante de integración:

$\begin{cases} \frac{\sqrt{3} \left(\frac{5 x^{3} \left(x^{3} - 1\right)^{\frac{3}{5}}}{3} - 12 \left(-1\right)^{\frac{3}{5}} x^{2} {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)} - 8 \left(-1\right)^{\frac{3}{5}} x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)} + \frac{25 \left(x^{3} - 1\right)^{\frac{3}{5}}}{9}\right) \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}{16 \pi} & \text{for}\: \left|{x^{3}}\right| > 1 \\\frac{\left(-1\right)^{\frac{3}{5}} \sqrt{3} \left(\frac{5 x^{3} \left(1 - x^{3}\right)^{\frac{3}{5}}}{3} - 12 x^{2} {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)} - 8 x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)} + \frac{25 \left(1 - x^{3}\right)^{\frac{3}{5}}}{9}\right) \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}{16 \pi} & \text{otherwese} \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} \frac{\sqrt{3} \left(\frac{5 x^{3} \left(x^{3} - 1\right)^{\frac{3}{5}}}{3} - 12 \left(-1\right)^{\frac{3}{5}} x^{2} {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)} - 8 \left(-1\right)^{\frac{3}{5}} x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)} + \frac{25 \left(x^{3} - 1\right)^{\frac{3}{5}}}{9}\right) \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}{16 \pi} & \text{for}\: \left|{x^{3}}\right| > 1 \\\frac{\left(-1\right)^{\frac{3}{5}} \sqrt{3} \left(\frac{5 x^{3} \left(1 - x^{3}\right)^{\frac{3}{5}}}{3} - 12 x^{2} {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)} - 8 x {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)} + \frac{25 \left(1 - x^{3}\right)^{\frac{3}{5}}}{9}\right) \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}{16 \pi} & \text{otherwese} \end{cases}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                                                                                                                    //                     3/5                 3/5                       \
                             -2*pi*I                                      -2*pi*I                                   ||            /      3\         3 /      3\                          |
  /                          -------              _                       -------              _                    ||         25*\-1 + x /      5*x *\-1 + x /                  | 3|    |
 |                        2     5                |_  /2/5, 2/3 |  3\         5                |_  /1/3, 2/5 |  3\   ||         --------------- + -----------------           for |x | > 1|
 |  5                    x *e       *Gamma(2/3)* |   |         | x |   x*e       *Gamma(1/3)* |   |         | x |   ||                72                 24                              |
 | x  + 3*x + 1                                 2  1 \  5/3    |   /                         2  1 \  4/3    |   /   ||                                                                   |
 | ------------ dx = C + ------------------------------------------- + ------------------------------------------ + |<                  -2*pi*I                     -2*pi*I              |
 |         2/5                            Gamma(5/3)                                  3*Gamma(4/3)                  ||             3/5  -------                3/5  -------              |
 | / 3    \                                                                                                         ||     /     3\        5         3 /     3\        5                 |
 | \x  - 1/                                                                                                         ||  25*\1 - x /   *e          5*x *\1 - x /   *e                     |
 |                                                                                                                  ||- ----------------------- - -------------------------   otherwise  |
/                                                                                                                   ||             72                         24                         |
                                                                                                                    \\                                                                   /

$$\int \frac{\left(x^{5} + 3 x\right) + 1}{\left(x^{3} - 1\right)^{\frac{2}{5}}}\, dx = C + \frac{x^{2} e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} + \frac{x e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \begin{cases} \frac{5 x^{3} \left(x^{3} - 1\right)^{\frac{3}{5}}}{24} + \frac{25 \left(x^{3} - 1\right)^{\frac{3}{5}}}{72} & \text{for}\: \left|{x^{3}}\right| > 1 \\- \frac{5 x^{3} \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{24} - \frac{25 \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{72} & \text{otherwise} \end{cases}$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

                                                                                                                                                                              3                                                                            
                                                                                                                                                                              /                                                                            
                                                                                                                                                                             |                                                                             
                                                                                                                                                                             |  /                                                    3/5                   
                                                                                                                                                                             |  |            5                2           2 /      3\                      
                                                                                                                                                                             |  |         3*x              5*x         5*x *\-1 + x /              3       
 -2*pi*I                                    -2*pi*I                                     -2*pi*I                                    -2*pi*I                                   |  |    -------------- + -------------- + -----------------      for x  > 1   
 -------              _                     -------              _                      -------              _                     -------              _                    |  |               2/5              2/5           8                           
    5                |_  /1/3, 2/5 |   \       5                |_  /2/5, 2/3 |  \         5                |_  /2/5, 2/3 |   \       5                |_  /1/3, 2/5 |  \    |  |      /      3\        /      3\                                          
e       *Gamma(1/3)* |   |         | 27|   e       *Gamma(2/3)* |   |         | 1|   9*e       *Gamma(2/3)* |   |         | 27|   e       *Gamma(1/3)* |   |         | 1|    |  |    8*\-1 + x /      8*\-1 + x /                                          
                    2  1 \  4/3    |   /                       2  1 \  5/3    |  /                         2  1 \  5/3    |   /                       2  1 \  4/3    |  /    |  |                                                                          
---------------------------------------- - --------------------------------------- + ------------------------------------------ - --------------------------------------- +  |  <                    -2*pi*I         -2*pi*I         -2*pi*I             dx
               Gamma(4/3)                                 Gamma(5/3)                                 Gamma(5/3)                                 3*Gamma(4/3)                 |  |               3/5  -------         -------         -------               
                                                                                                                                                                             |  |     2 /     3\        5         5     5         2     5                  
                                                                                                                                                                             |  |  5*x *\1 - x /   *e          3*x *e          5*x *e                      
                                                                                                                                                                             |  |- ------------------------- + ------------- + -------------  otherwise    
                                                                                                                                                                             |  |              8                         2/5             2/5               
                                                                                                                                                                             |  |                                /     3\        /     3\                  
                                                                                                                                                                             |  |                              8*\1 - x /      8*\1 - x /                  
                                                                                                                                                                             |  \                                                                          
                                                                                                                                                                             |                                                                             
                                                                                                                                                                            /                                                                              
                                                                                                                                                                            1

$$- \frac{e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {1} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{5} \\ \frac{4}{3} \end{matrix}\middle| {27} \right)}}{\Gamma\left(\frac{4}{3}\right)} - \frac{e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {1} \right)}}{\Gamma\left(\frac{5}{3}\right)} + \frac{9 e^{- \frac{2 i \pi}{5}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{2}{5}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {27} \right)}}{\Gamma\left(\frac{5}{3}\right)} + \int\limits_{1}^{3} \begin{cases} \frac{3 x^{5}}{8 \left(x^{3} - 1\right)^{\frac{2}{5}}} + \frac{5 x^{2} \left(x^{3} - 1\right)^{\frac{3}{5}}}{8} + \frac{5 x^{2}}{8 \left(x^{3} - 1\right)^{\frac{2}{5}}} & \text{for}\: x^{3} > 1 \\\frac{3 x^{5} e^{- \frac{2 i \pi}{5}}}{8 \left(1 - x^{3}\right)^{\frac{2}{5}}} - \frac{5 x^{2} \left(1 - x^{3}\right)^{\frac{3}{5}} e^{- \frac{2 i \pi}{5}}}{8} + \frac{5 x^{2} e^{- \frac{2 i \pi}{5}}}{8 \left(1 - x^{3}\right)^{\frac{2}{5}}} & \text{otherwise} \end{cases}\, dx$$

                                                                                                                                                                              3                                                                            
                                                                                                                                                                              /                                                                            
                                                                                                                                                                             |                                                                             
                                                                                                                                                                             |  /                                                    3/5                   
                                                                                                                                                                             |  |            5                2           2 /      3\                      
                                                                                                                                                                             |  |         3*x              5*x         5*x *\-1 + x /              3       
 -2*pi*I                                    -2*pi*I                                     -2*pi*I                                    -2*pi*I                                   |  |    -------------- + -------------- + -----------------      for x  > 1   
 -------              _                     -------              _                      -------              _                     -------              _                    |  |               2/5              2/5           8                           
    5                |_  /1/3, 2/5 |   \       5                |_  /2/5, 2/3 |  \         5                |_  /2/5, 2/3 |   \       5                |_  /1/3, 2/5 |  \    |  |      /      3\        /      3\                                          
e       *Gamma(1/3)* |   |         | 27|   e       *Gamma(2/3)* |   |         | 1|   9*e       *Gamma(2/3)* |   |         | 27|   e       *Gamma(1/3)* |   |         | 1|    |  |    8*\-1 + x /      8*\-1 + x /                                          
                    2  1 \  4/3    |   /                       2  1 \  5/3    |  /                         2  1 \  5/3    |   /                       2  1 \  4/3    |  /    |  |                                                                          
---------------------------------------- - --------------------------------------- + ------------------------------------------ - --------------------------------------- +  |  <                    -2*pi*I         -2*pi*I         -2*pi*I             dx
               Gamma(4/3)                                 Gamma(5/3)                                 Gamma(5/3)                                 3*Gamma(4/3)                 |  |               3/5  -------         -------         -------               
                                                                                                                                                                             |  |     2 /     3\        5         5     5         2     5                  
                                                                                                                                                                             |  |  5*x *\1 - x /   *e          3*x *e          5*x *e                      
                                                                                                                                                                             |  |- ------------------------- + ------------- + -------------  otherwise    
                                                                                                                                                                             |  |              8                         2/5             2/5               
                                                                                                                                                                             |  |                                /     3\        /     3\                  
                                                                                                                                                                             |  |                              8*\1 - x /      8*\1 - x /                  
                                                                                                                                                                             |  \                                                                          
                                                                                                                                                                             |                                                                             
                                                                                                                                                                            /                                                                              
                                                                                                                                                                            1

exp(-2*pi*i/5)*gamma(1/3)*hyper((1/3, 2/5), (4/3,), 27)/gamma(4/3) - exp(-2*pi*i/5)*gamma(2/3)*hyper((2/5, 2/3), (5/3,), 1)/gamma(5/3) + 9*exp(-2*pi*i/5)*gamma(2/3)*hyper((2/5, 2/3), (5/3,), 27)/gamma(5/3) - exp(-2*pi*i/5)*gamma(1/3)*hyper((1/3, 2/5), (4/3,), 1)/(3*gamma(4/3)) + Integral(Piecewise((3*x^5/(8*(-1 + x^3)^(2/5)) + 5*x^2/(8*(-1 + x^3)^(2/5)) + 5*x^2*(-1 + x^3)^(3/5)/8, x^3 > 1), (-5*x^2*(1 - x^3)^(3/5)*exp(-2*pi*i/5)/8 + 3*x^5*exp(-2*pi*i/5)/(8*(1 - x^3)^(2/5)) + 5*x^2*exp(-2*pi*i/5)/(8*(1 - x^3)^(2/5)), True)), (x, 1, 3))

Respuesta numérica [src]

49.8533586757794

49.8533586757794

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

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Expresiones idénticas

Expresiones semejantes

Integral de (x^5+3x+1)/(x^3-1)^(2/5) dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución