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

Integral de sqr(x)*cbrt((3-x^3)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   ___                     
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  |                    2   
  |         ___________    
  |        /         2     
  |     3 /  /     3\      
  |   x*\/   \3 - x /    dx
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 1
∫13x((3−x3)23)2 dx\int\limits_{1}^{\sqrt{3}} x \left(\sqrt[3]{\left(3 - x^{3}\right)^{2}}\right)^{2}\, dx 
Integral(x*(((3 - x^3)^2)^(1/3))^2, (x, 1, sqrt(3)))
Respuesta [src] 
   ___                                                                                                                                      
 \/ 3                                                                                                                                       
   /                                                                                                                                        
  |                                                                                                                                         
  |   /              -2*pi*I                                                -2*pi*I                                                         
  |   |              -------              _  /          |  3\               -------              _  /          |  3\                        
  |   |       3 ___     3                |_  |-4/3, 2/3 | x |     3 ___  4     3                |_  |-1/3, 5/3 | x |                        
  |   |   2*x*\/ 3 *e       *Gamma(2/3)* |   |          | --|   8*\/ 3 *x *e       *Gamma(2/3)* |   |          | --|                        
  |   |                                 2  1 \   5/3    | 3 /                                  2  1 \   8/3    | 3 /               3        
  |   |   --------------------------------------------------- - ----------------------------------------------------     for -3 + x  >= 0   
  |   |                        Gamma(5/3)                                          15*Gamma(5/3)                                            
  |   <                                                                                                                                   dx
  |   |                                                                                                                                     
  |   |                       _  /          |  3  2*pi*I\                                   _  /          |  3  2*pi*I\                     
  |   |    3 ___             |_  |-4/3, 2/3 | x *e      |     3 ___  4  2*pi*I             |_  |-1/3, 5/3 | x *e      |                     
  |   |2*x*\/ 3 *Gamma(2/3)* |   |          | ----------|   8*\/ 3 *x *e      *Gamma(2/3)* |   |          | ----------|                     
  |   |                     2  1 \   5/3    |     3     /                                 2  1 \   8/3    |     3     /                     
  |   |-------------------------------------------------- - -----------------------------------------------------------     otherwise       
  |   \                    Gamma(5/3)                                              15*Gamma(5/3)                                            
  |                                                                                                                                         
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 1
∫13{−833x4e−2iπ3Γ(23)2F1(−13,5383|x33)15Γ(53)+233xe−2iπ3Γ(23)2F1(−43,2353|x33)Γ(53)for x3−3≥0−833x4e2iπΓ(23)2F1(−13,5383|x3e2iπ3)15Γ(53)+233xΓ(23)2F1(−43,2353|x3e2iπ3)Γ(53)otherwise dx\int\limits_{1}^{\sqrt{3}} \begin{cases} - \frac{8 \sqrt[3]{3} x^{4} 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| {\frac{x^{3}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x e^{- \frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{for}\: x^{3} - 3 \geq 0 \\- \frac{8 \sqrt[3]{3} x^{4} 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| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{otherwise} \end{cases}\, dx 
=
= 
   ___                                                                                                                                      
 \/ 3                                                                                                                                       
   /                                                                                                                                        
  |                                                                                                                                         
  |   /              -2*pi*I                                                -2*pi*I                                                         
  |   |              -------              _  /          |  3\               -------              _  /          |  3\                        
  |   |       3 ___     3                |_  |-4/3, 2/3 | x |     3 ___  4     3                |_  |-1/3, 5/3 | x |                        
  |   |   2*x*\/ 3 *e       *Gamma(2/3)* |   |          | --|   8*\/ 3 *x *e       *Gamma(2/3)* |   |          | --|                        
  |   |                                 2  1 \   5/3    | 3 /                                  2  1 \   8/3    | 3 /               3        
  |   |   --------------------------------------------------- - ----------------------------------------------------     for -3 + x  >= 0   
  |   |                        Gamma(5/3)                                          15*Gamma(5/3)                                            
  |   <                                                                                                                                   dx
  |   |                                                                                                                                     
  |   |                       _  /          |  3  2*pi*I\                                   _  /          |  3  2*pi*I\                     
  |   |    3 ___             |_  |-4/3, 2/3 | x *e      |     3 ___  4  2*pi*I             |_  |-1/3, 5/3 | x *e      |                     
  |   |2*x*\/ 3 *Gamma(2/3)* |   |          | ----------|   8*\/ 3 *x *e      *Gamma(2/3)* |   |          | ----------|                     
  |   |                     2  1 \   5/3    |     3     /                                 2  1 \   8/3    |     3     /                     
  |   |-------------------------------------------------- - -----------------------------------------------------------     otherwise       
  |   \                    Gamma(5/3)                                              15*Gamma(5/3)                                            
  |                                                                                                                                         
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 1
∫13{−833x4e−2iπ3Γ(23)2F1(−13,5383|x33)15Γ(53)+233xe−2iπ3Γ(23)2F1(−43,2353|x33)Γ(53)for x3−3≥0−833x4e2iπΓ(23)2F1(−13,5383|x3e2iπ3)15Γ(53)+233xΓ(23)2F1(−43,2353|x3e2iπ3)Γ(53)otherwise dx\int\limits_{1}^{\sqrt{3}} \begin{cases} - \frac{8 \sqrt[3]{3} x^{4} 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| {\frac{x^{3}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x e^{- \frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{for}\: x^{3} - 3 \geq 0 \\- \frac{8 \sqrt[3]{3} x^{4} 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| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((2*x*3^(1/3)*exp(-2*pi*i/3)*gamma(2/3)*hyper((-4/3, 2/3), (5/3,), x^3/3)/gamma(5/3) - 8*3^(1/3)*x^4*exp(-2*pi*i/3)*gamma(2/3)*hyper((-1/3, 5/3), (8/3,), x^3/3)/(15*gamma(5/3)), -3 + x^3 >= 0), (2*x*3^(1/3)*gamma(2/3)*hyper((-4/3, 2/3), (5/3,), x^3*exp_polar(2*pi*i)/3)/gamma(5/3) - 8*3^(1/3)*x^4*exp_polar(2*pi*i)*gamma(2/3)*hyper((-1/3, 5/3), (8/3,), x^3*exp_polar(2*pi*i)/3)/(15*gamma(5/3)), True)), (x, 1, sqrt(3)))
Respuesta numérica [src] 
1.16792409981634
1.16792409981634

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

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