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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

Solución

Ha introducido [src]

  1                  
  /                  
 |                   
 |  1 + log(x + 1)   
 |  -------------- dx
 |      x - 1        
 |                   
/                    
0

$$\int\limits_{0}^{1} \frac{\log{\left(x + 1 \right)} + 1}{x - 1}\, dx$$

Integral((1 + log(x + 1))/(x - 1), (x, 0, 1))

Solución detallada

Vuelva a escribir el integrando:

$\frac{\log{\left(x + 1 \right)} + 1}{x - 1} = \frac{\log{\left(x + 1 \right)}}{x - 1} + \frac{1}{x - 1}$
Integramos término a término:
1. No puedo encontrar los pasos en la búsqueda de esta integral.
  
  Pero la integral
  
  $\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases}$
1. que $u = x - 1$ .
  
  Luego que $du = dx$ y ponemos $du$ :
  
  $\int \frac{1}{u}\, du$
  1. Integral $\frac{1}{u}$ es $\log{\left(u \right)}$ .
  Si ahora sustituir $u$ más en:
  
  $\log{\left(x - 1 \right)}$
El resultado es: $\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases} + \log{\left(x - 1 \right)}$
Ahora simplificar:

$\begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwese} \end{cases}$
Añadimos la constante de integración:

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

Respuesta:

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

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       |       /                  |             
                           \\                                                                                                                            /

$$\int \frac{\log{\left(x + 1 \right)} + 1}{x - 1}\, dx = C + \begin{cases} \log{\left(2 \right)} \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \left|{x - 1}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x - 1} \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x - 1}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x - 1} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{\left(x - 1\right) e^{i \pi}}{2}\right) & \text{otherwise} \end{cases} + \log{\left(x - 1 \right)}$$

Simplificar

Respuesta [src]

-oo - pi*I - pi*I*log(2)

$$-\infty - i \pi - i \pi \log{\left(2 \right)}$$

-oo - pi*I - pi*I*log(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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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

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

Límites de integración:

Gráfico:

Definida a trozos:

Solución