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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

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