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Resolución de integrales/ x^(2*n)/ x^n/(1+x^(2*n))

Integral de x^n/(1+x^(2*n)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |      n      
 |     x       
 |  -------- dx
 |       2*n   
 |  1 + x      
 |             
/              
0
$$\int\limits_{0}^{1} \frac{x^{n}}{x^{2 n} + 1}\, dx$$ 
Integral(x^n/(1 + x^(2*n)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                             
 |                      n      /1    1 \         / 2*n  pi*I     1    1 \      n      /1    1 \         / 2*n  pi*I     1    1 \
 |     n             x*x *Gamma|- + ---|*lerchphi|x   *e    , 1, - + ---|   x*x *Gamma|- + ---|*lerchphi|x   *e    , 1, - + ---|
 |    x                        \2   2*n/         \               2   2*n/             \2   2*n/         \               2   2*n/
 | -------- dx = C + ---------------------------------------------------- + ----------------------------------------------------
 |      2*n                                    /3    1 \                                       2      /3    1 \                 
 | 1 + x                              4*n*Gamma|- + ---|                                    4*n *Gamma|- + ---|                 
 |                                             \2   2*n/                                              \2   2*n/                 
/
$$\int \frac{x^{n}}{x^{2 n} + 1}\, dx = C + \frac{x x^{n} \Phi\left(x^{2 n} e^{i \pi}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)} + \frac{x x^{n} \Phi\left(x^{2 n} e^{i \pi}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n^{2} \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)}$$
Simplificar
Respuesta [src] 
     /1    1 \         / pi*I     1    1 \        /1    1 \         / pi*I     1    1 \
Gamma|- + ---|*lerchphi|e    , 1, - + ---|   Gamma|- + ---|*lerchphi|e    , 1, - + ---|
     \2   2*n/         \          2   2*n/        \2   2*n/         \          2   2*n/
------------------------------------------ + ------------------------------------------
                     /3    1 \                             2      /3    1 \            
            4*n*Gamma|- + ---|                          4*n *Gamma|- + ---|            
                     \2   2*n/                                    \2   2*n/
$$\frac{\Phi\left(e^{i \pi}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)} + \frac{\Phi\left(e^{i \pi}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n^{2} \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)}$$ 
=
= 
     /1    1 \         / pi*I     1    1 \        /1    1 \         / pi*I     1    1 \
Gamma|- + ---|*lerchphi|e    , 1, - + ---|   Gamma|- + ---|*lerchphi|e    , 1, - + ---|
     \2   2*n/         \          2   2*n/        \2   2*n/         \          2   2*n/
------------------------------------------ + ------------------------------------------
                     /3    1 \                             2      /3    1 \            
            4*n*Gamma|- + ---|                          4*n *Gamma|- + ---|            
                     \2   2*n/                                    \2   2*n/
$$\frac{\Phi\left(e^{i \pi}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)} + \frac{\Phi\left(e^{i \pi}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n^{2} \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)}$$ 
gamma(1/2 + 1/(2*n))*lerchphi(exp_polar(pi*i), 1, 1/2 + 1/(2*n))/(4*n*gamma(3/2 + 1/(2*n))) + gamma(1/2 + 1/(2*n))*lerchphi(exp_polar(pi*i), 1, 1/2 + 1/(2*n))/(4*n^2*gamma(3/2 + 1/(2*n)))

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

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