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

Integral de x^(1÷3)/(1+sqrt(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1             
  /             
 |              
 |    3 ___     
 |    \/ x      
 |  --------- dx
 |        ___   
 |  1 + \/ x    
 |              
/               
0
∫01x3x+1 dx\int\limits_{0}^{1} \frac{\sqrt[3]{x}}{\sqrt{x} + 1}\, dx 
Integral(x^(1/3)/(1 + sqrt(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                      -2*pi*I               /           pi*I\       2*pi*I               /           5*pi*I\
 |                                                                                                       -------               |           ----|       ------               |           ------|
 |   3 ___              3 ___                               /    6 ___  pi*I\       5/6                     3                  |    6 ___   3  |         3                  |    6 ___    3   |
 |   \/ x             8*\/ x *Gamma(8/3)   16*Gamma(8/3)*log\1 - \/ x *e    /   16*x   *Gamma(8/3)   16*e       *Gamma(8/3)*log\1 - \/ x *e    /   16*e      *Gamma(8/3)*log\1 - \/ x *e      /
 | --------- dx = C - ------------------ - ---------------------------------- + ------------------ - ------------------------------------------- - --------------------------------------------
 |       ___             Gamma(11/3)                 3*Gamma(11/3)                5*Gamma(11/3)                     3*Gamma(11/3)                                 3*Gamma(11/3)                
 | 1 + \/ x                                                                                                                                                                                    
 |                                                                                                                                                                                             
/
∫x3x+1 dx=C+16x56Γ(83)5Γ(113)−8x3Γ(83)Γ(113)−16e−2iπ3log⁡(−x6eiπ3+1)Γ(83)3Γ(113)−16log⁡(−x6eiπ+1)Γ(83)3Γ(113)−16e2iπ3log⁡(−x6e5iπ3+1)Γ(83)3Γ(113)\int \frac{\sqrt[3]{x}}{\sqrt{x} + 1}\, dx = C + \frac{16 x^{\frac{5}{6}} \Gamma\left(\frac{8}{3}\right)}{5 \Gamma\left(\frac{11}{3}\right)} - \frac{8 \sqrt[3]{x} \Gamma\left(\frac{8}{3}\right)}{\Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{- \frac{2 i \pi}{3}} \log{\left(- \sqrt[6]{x} e^{\frac{i \pi}{3}} + 1 \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 \log{\left(- \sqrt[6]{x} e^{i \pi} + 1 \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{\frac{2 i \pi}{3}} \log{\left(- \sqrt[6]{x} e^{\frac{5 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)}
Simplificar
Respuesta [src] 
                                                     -2*pi*I               /     pi*I\       2*pi*I               /     5*pi*I\
                                                     -------               |     ----|       ------               |     ------|
                                   /     pi*I\          3                  |      3  |         3                  |       3   |
  24*Gamma(8/3)   16*Gamma(8/3)*log\1 - e    /   16*e       *Gamma(8/3)*log\1 - e    /   16*e      *Gamma(8/3)*log\1 - e      /
- ------------- - ---------------------------- - ------------------------------------- - --------------------------------------
  5*Gamma(11/3)          3*Gamma(11/3)                       3*Gamma(11/3)                           3*Gamma(11/3)
−16e2iπ3log⁡(1−e5iπ3)Γ(83)3Γ(113)−24Γ(83)5Γ(113)−16log⁡(1−eiπ)Γ(83)3Γ(113)−16e−2iπ3log⁡(1−eiπ3)Γ(83)3Γ(113)- \frac{16 e^{\frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{5 i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{24 \Gamma\left(\frac{8}{3}\right)}{5 \Gamma\left(\frac{11}{3}\right)} - \frac{16 \log{\left(1 - e^{i \pi} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{- \frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} 
=
= 
                                                     -2*pi*I               /     pi*I\       2*pi*I               /     5*pi*I\
                                                     -------               |     ----|       ------               |     ------|
                                   /     pi*I\          3                  |      3  |         3                  |       3   |
  24*Gamma(8/3)   16*Gamma(8/3)*log\1 - e    /   16*e       *Gamma(8/3)*log\1 - e    /   16*e      *Gamma(8/3)*log\1 - e      /
- ------------- - ---------------------------- - ------------------------------------- - --------------------------------------
  5*Gamma(11/3)          3*Gamma(11/3)                       3*Gamma(11/3)                           3*Gamma(11/3)
−16e2iπ3log⁡(1−e5iπ3)Γ(83)3Γ(113)−24Γ(83)5Γ(113)−16log⁡(1−eiπ)Γ(83)3Γ(113)−16e−2iπ3log⁡(1−eiπ3)Γ(83)3Γ(113)- \frac{16 e^{\frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{5 i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{24 \Gamma\left(\frac{8}{3}\right)}{5 \Gamma\left(\frac{11}{3}\right)} - \frac{16 \log{\left(1 - e^{i \pi} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{- \frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} 
-24*gamma(8/3)/(5*gamma(11/3)) - 16*gamma(8/3)*log(1 - exp_polar(pi*i))/(3*gamma(11/3)) - 16*exp(-2*pi*i/3)*gamma(8/3)*log(1 - exp_polar(pi*i/3))/(3*gamma(11/3)) - 16*exp(2*pi*i/3)*gamma(8/3)*log(1 - exp_polar(5*pi*i/3))/(3*gamma(11/3))
Respuesta numérica [src] 
0.441304367348545
0.441304367348545

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

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