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Resolución de integrales/ lnx/x/ lnx/x^n

Integral de lnx/x^n dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo          
  /          
 |           
 |  log(x)   
 |  ------ dx
 |     n     
 |    x      
 |           
/            
1
∫1∞log⁡(x)xn dx\int\limits_{1}^{\infty} \frac{\log{\left(x \right)}}{x^{n}}\, dx 
Integral(log(x)/x^n, (x, 1, oo))
Solución detallada

  1. Usamos la integración por partes:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    que u(x)=log⁡(x)u{\left(x \right)} = \log{\left(x \right)} y que dv⁡(x)=x−n\operatorname{dv}{\left(x \right)} = x^{- n}.

    Entonces du⁡(x)=1x\operatorname{du}{\left(x \right)} = \frac{1}{x}.

    Para buscar v(x)v{\left(x \right)}:

    1. Integral xnx^{n} es xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x−n dx={x1−n1−nfor n≠1log⁡(x)otherwese\int x^{- n}\, dx = \begin{cases} \frac{x^{1 - n}}{1 - n} & \text{for}\: n \neq 1 \\\log{\left(x \right)} & \text{otherwese} \end{cases}

    Ahora resolvemos podintegral.

  2. No puedo encontrar los pasos en la búsqueda de esta integral.

    Pero la integral

    {{−xnxn−xnfor n≠1log⁡(x)otherwese1−nfor n>−∞∧n<∞∧n≠1log⁡(x)22otherwese\begin{cases} \frac{\begin{cases} - \frac{x}{n x^{n} - x^{n}} & \text{for}\: n \neq 1 \\\log{\left(x \right)} & \text{otherwese} \end{cases}}{1 - n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 1 \\\frac{\log{\left(x \right)}^{2}}{2} & \text{otherwese} \end{cases}

  3. Ahora simplificar:

    {x1−n(−n−(n−1)2log⁡(x)+1)(n−1)3for n>−∞∧n<∞∧n≠1(−2x1−n+(1−n)log⁡(x))log⁡(x)2(n−1)for n≠1log⁡(x)22otherwese\begin{cases} \frac{x^{1 - n} \left(- n - \left(n - 1\right)^{2} \log{\left(x \right)} + 1\right)}{\left(n - 1\right)^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 1 \\\frac{\left(- 2 x^{1 - n} + \left(1 - n\right) \log{\left(x \right)}\right) \log{\left(x \right)}}{2 \left(n - 1\right)} & \text{for}\: n \neq 1 \\\frac{\log{\left(x \right)}^{2}}{2} & \text{otherwese} \end{cases}

  4. Añadimos la constante de integración:

    {x1−n(−n−(n−1)2log⁡(x)+1)(n−1)3for n>−∞∧n<∞∧n≠1(−2x1−n+(1−n)log⁡(x))log⁡(x)2(n−1)for n≠1log⁡(x)22otherwese+constant\begin{cases} \frac{x^{1 - n} \left(- n - \left(n - 1\right)^{2} \log{\left(x \right)} + 1\right)}{\left(n - 1\right)^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 1 \\\frac{\left(- 2 x^{1 - n} + \left(1 - n\right) \log{\left(x \right)}\right) \log{\left(x \right)}}{2 \left(n - 1\right)} & \text{for}\: n \neq 1 \\\frac{\log{\left(x \right)}^{2}}{2} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta:

{x1−n(−n−(n−1)2log⁡(x)+1)(n−1)3for n>−∞∧n<∞∧n≠1(−2x1−n+(1−n)log⁡(x))log⁡(x)2(n−1)for n≠1log⁡(x)22otherwese+constant\begin{cases} \frac{x^{1 - n} \left(- n - \left(n - 1\right)^{2} \log{\left(x \right)} + 1\right)}{\left(n - 1\right)^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 1 \\\frac{\left(- 2 x^{1 - n} + \left(1 - n\right) \log{\left(x \right)}\right) \log{\left(x \right)}}{2 \left(n - 1\right)} & \text{for}\: n \neq 1 \\\frac{\log{\left(x \right)}^{2}}{2} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                   ///    -x                                                   \                               
                   |||-----------  for n != 1                                  |                               
                   |||   n      n                                              |                               
                   ||<- x  + n*x                                               |                               
  /                |||                                                         |   // 1 - n            \       
 |                 |||  log(x)     otherwise                                   |   ||x                 |       
 | log(x)          ||\                                                         |   ||------  for n != 1|       
 | ------ dx = C - |<------------------------  for And(n > -oo, n < oo, n != 1)| + |<1 - n             |*log(x)
 |    n            ||         1 - n                                            |   ||                  |       
 |   x             ||                                                          |   ||log(x)  otherwise |       
 |                 ||           2                                              |   \\                  /       
/                  ||        log (x)                                           |                               
                   ||        -------                      otherwise            |                               
                   ||           2                                              |                               
                   \\                                                          /
∫log⁡(x)xn dx=C+({x1−n1−nfor n≠1log⁡(x)otherwise)log⁡(x)−{{−xnxn−xnfor n≠1log⁡(x)otherwise1−nfor n>−∞∧n<∞∧n≠1log⁡(x)22otherwise\int \frac{\log{\left(x \right)}}{x^{n}}\, dx = C + \left(\begin{cases} \frac{x^{1 - n}}{1 - n} & \text{for}\: n \neq 1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}\right) \log{\left(x \right)} - \begin{cases} \frac{\begin{cases} - \frac{x}{n x^{n} - x^{n}} & \text{for}\: n \neq 1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}}{1 - n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 1 \\\frac{\log{\left(x \right)}^{2}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/        1                       
|    ---------      for re(n) > 1
|            2                   
|    (-1 + n)                    
|                                
| oo                             
<  /                             
| |                              
| |   -n                         
| |  x  *log(x) dx    otherwise  
| |                              
|/                               
\1
{1(n−1)2for re⁡(n)>1∫1∞x−nlog⁡(x) dxotherwise\begin{cases} \frac{1}{\left(n - 1\right)^{2}} & \text{for}\: \operatorname{re}{\left(n\right)} > 1 \\\int\limits_{1}^{\infty} x^{- n} \log{\left(x \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/        1                       
|    ---------      for re(n) > 1
|            2                   
|    (-1 + n)                    
|                                
| oo                             
<  /                             
| |                              
| |   -n                         
| |  x  *log(x) dx    otherwise  
| |                              
|/                               
\1
{1(n−1)2for re⁡(n)>1∫1∞x−nlog⁡(x) dxotherwise\begin{cases} \frac{1}{\left(n - 1\right)^{2}} & \text{for}\: \operatorname{re}{\left(n\right)} > 1 \\\int\limits_{1}^{\infty} x^{- n} \log{\left(x \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise(((-1 + n)^(-2), re(n) > 1), (Integral(x^(-n)*log(x), (x, 1, oo)), True))

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

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