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Resolución de integrales/ (x+1)/ (x-1)/ (1+ln(x+1))/(x-1)

Integral de (1+ln(x+1))/(x-1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |  1 + log(x + 1)   
 |  -------------- dx
 |      x - 1        
 |                   
/                    
0
∫01log⁡(x+1)+1x−1 dx\int\limits_{0}^{1} \frac{\log{\left(x + 1 \right)} + 1}{x - 1}\, dx 
Integral((1 + log(x + 1))/(x - 1), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    log⁡(x+1)+1x−1=log⁡(x+1)x−1+1x−1\frac{\log{\left(x + 1 \right)} + 1}{x - 1} = \frac{\log{\left(x + 1 \right)}}{x - 1} + \frac{1}{x - 1}

  2. Integramos término a término:

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

      Pero la integral

      {log⁡(2)log⁡(x−1)−Li⁡2((x−1)eiπ2)for ∣x−1∣<1−log⁡(2)log⁡(1x−1)−Li⁡2((x−1)eiπ2)for 1∣x−1∣<1−G2,22,0(1,10,0|x−1)log⁡(2)+G2,20,2(1,10,0|x−1)log⁡(2)−Li⁡2((x−1)eiπ2)otherwese\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases}

    1. que u=x−1u = x - 1.

      Luego que du=dxdu = dx y ponemos dudu:

      ∫1u du\int \frac{1}{u}\, du

      1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

      Si ahora sustituir uu más en:

      log⁡(x−1)\log{\left(x - 1 \right)}

    El resultado es: {log⁡(2)log⁡(x−1)−Li⁡2((x−1)eiπ2)for ∣x−1∣<1−log⁡(2)log⁡(1x−1)−Li⁡2((x−1)eiπ2)for 1∣x−1∣<1−G2,22,0(1,10,0|x−1)log⁡(2)+G2,20,2(1,10,0|x−1)log⁡(2)−Li⁡2((x−1)eiπ2)otherwese+log⁡(x−1)\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases} + \log{\left(x - 1 \right)}

  3. Ahora simplificar:

    {log⁡(2)log⁡(x−1)+log⁡(x−1)−Li⁡2((x−1)eiπ2)for ∣x−1∣<1−log⁡(2)log⁡(1x−1)+log⁡(x−1)−Li⁡2((x−1)eiπ2)for 1∣x−1∣<1−G2,22,0(1,10,0|x−1)log⁡(2)+G2,20,2(1,10,0|x−1)log⁡(2)+log⁡(x−1)−Li⁡2((x−1)eiπ2)otherwese\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases}

  4. Añadimos la constante de integración:

    {log⁡(2)log⁡(x−1)+log⁡(x−1)−Li⁡2((x−1)eiπ2)for ∣x−1∣<1−log⁡(2)log⁡(1x−1)+log⁡(x−1)−Li⁡2((x−1)eiπ2)for 1∣x−1∣<1−G2,22,0(1,10,0|x−1)log⁡(2)+G2,20,2(1,10,0|x−1)log⁡(2)+log⁡(x−1)−Li⁡2((x−1)eiπ2)otherwese+constant\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta:

{log⁡(2)log⁡(x−1)+log⁡(x−1)−Li⁡2((x−1)eiπ2)for ∣x−1∣<1−log⁡(2)log⁡(1x−1)+log⁡(x−1)−Li⁡2((x−1)eiπ2)for 1∣x−1∣<1−G2,22,0(1,10,0|x−1)log⁡(2)+G2,20,2(1,10,0|x−1)log⁡(2)+log⁡(x−1)−Li⁡2((x−1)eiπ2)otherwese+constant\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                           //                                     /             pi*I\                                                                    \             
                           ||                                     |   (-1 + x)*e    |                                                                    |             
                           ||                            - polylog|2, --------------| + log(2)*log(-1 + x)                               for |-1 + x| < 1|             
                           ||                                     \         2       /                                                                    |             
  /                        ||                                                                                                                            |             
 |                         ||                                     /             pi*I\                                                                    |             
 | 1 + log(x + 1)          ||                                     |   (-1 + x)*e    |             /  1   \                                      1        |             
 | -------------- dx = C + |<                            - polylog|2, --------------| - log(2)*log|------|                               for -------- < 1| + log(x - 1)
 |     x - 1               ||                                     \         2       /             \-1 + x/                                   |-1 + x|    |             
 |                         ||                                                                                                                            |             
/                          ||         /             pi*I\                                                                                                |             
                           ||         |   (-1 + x)*e    |           __0, 2 /1, 1       |       \           __2, 0 /      1, 1 |       \                  |             
                           ||- polylog|2, --------------| + log(2)*/__     |           | -1 + x| - log(2)*/__     |           | -1 + x|     otherwise    |             
                           ||         \         2       /          \_|2, 2 \      0, 0 |       /          \_|2, 2 \0, 0       |       /                  |             
                           \\                                                                                                                            /
∫log⁡(x+1)+1x−1 dx=C+{log⁡(2)log⁡(x−1)−Li⁡2((x−1)eiπ2)for ∣x−1∣<1−log⁡(2)log⁡(1x−1)−Li⁡2((x−1)eiπ2)for 1∣x−1∣<1−G2,22,0(1,10,0|x−1)log⁡(2)+G2,20,2(1,10,0|x−1)log⁡(2)−Li⁡2((x−1)eiπ2)otherwise+log⁡(x−1)\int \frac{\log{\left(x + 1 \right)} + 1}{x - 1}\, dx = C + \begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwise} \end{cases} + \log{\left(x - 1 \right)}
Simplificar
Respuesta [src] 
-oo - pi*I - pi*I*log(2)
−∞−iπ−iπlog⁡(2)-\infty - i \pi - i \pi \log{\left(2 \right)} 
=
= 
-oo - pi*I - pi*I*log(2)
−∞−iπ−iπlog⁡(2)-\infty - i \pi - i \pi \log{\left(2 \right)} 
-oo - pi*i - pi*i*log(2)
Respuesta numérica [src] 
-74.0702386443129
-74.0702386443129

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

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