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Resolución de integrales/ x/(x²-1)/ lnx/(x²-1)

Integral de lnx/(x²-1) dx

Solución

Ha introducido [src]

  1          
  /          
 |           
 |  log(x)   
 |  ------ dx
 |   2       
 |  x  - 1   
 |           
/            
0

$$\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{x^{2} - 1}\, dx$$

Integral(log(x)/(x^2 - 1), (x, 0, 1))

Solución detallada

Usamos la integración por partes:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

que $u{\left(x \right)} = \log{\left(x \right)}$ y que $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} - 1}$ .

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

Para buscar $v{\left(x \right)}$ :
Ahora resolvemos podintegral.
No puedo encontrar los pasos en la búsqueda de esta integral.

Pero la integral

$\begin{cases} - \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx & \text{for}\: x < -1 \\- \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x < 1 \\- \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{otherwese} \end{cases}$
Ahora simplificar:

$\begin{cases} - \log{\left(x \right)} \operatorname{acoth}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx & \text{for}\: x < -1 \\- \log{\left(x \right)} \operatorname{atanh}{\left(x \right)} + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x > -1 \wedge x < 1 \\- \log{\left(x \right)} \operatorname{acoth}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x^{2} > 1 \\- \log{\left(x \right)} \operatorname{atanh}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x^{2} < 1 \end{cases}$
Añadimos la constante de integración:

$\begin{cases} - \log{\left(x \right)} \operatorname{acoth}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx & \text{for}\: x < -1 \\- \log{\left(x \right)} \operatorname{atanh}{\left(x \right)} + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x > -1 \wedge x < 1 \\- \log{\left(x \right)} \operatorname{acoth}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x^{2} > 1 \\- \log{\left(x \right)} \operatorname{atanh}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x^{2} < 1 \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} - \log{\left(x \right)} \operatorname{acoth}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx & \text{for}\: x < -1 \\- \log{\left(x \right)} \operatorname{atanh}{\left(x \right)} + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x > -1 \wedge x < 1 \\- \log{\left(x \right)} \operatorname{acoth}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x^{2} > 1 \\- \log{\left(x \right)} \operatorname{atanh}{\left(x \right)} + \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x^{2} < 1 \end{cases}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                   //                                       /                                                            \                                  
                   ||                                      |                                                             |                                  
                   ||                                      | acoth(x)                                                    |                                  
                   ||                                    - | -------- dx                                       for x < -1|                                  
                   ||                                      |    x                                                        |                                  
                   ||                                      |                                                             |                                  
                   ||                                     /                                                              |                                  
                   ||                                                                                                    |                                  
                   ||                     -1                                 -1                                          |                                  
                   ||                      /                 /                /                                          |                                  
  /                ||                     |                 |                |                                           |                                  
 |                 ||                     |  acoth(x)       | atanh(x)       |  atanh(x)                                 |   //                2    \       
 | log(x)          ||                  -  |  -------- dx -  | -------- dx +  |  -------- dx                    for x < 1 |   ||-acoth(x)  for x  > 1|       
 | ------ dx = C - |<                     |     x           |    x           |     x                                     | + |<                     |*log(x)
 |  2              ||                     |                 |                |                                           |   ||                2    |       
 | x  - 1          ||                    /                 /                /                                            |   \\-atanh(x)  for x  < 1/       
 |                 ||                                                                                                    |                                  
/                  ||                                                                                                    |                                  
                   ||                    -1                 1                 1                -1                        |                                  
                   ||    /                /                 /                 /                 /                        |                                  
                   ||   |                |                 |                 |                 |                         |                                  
                   ||   | acoth(x)       |  acoth(x)       |  atanh(x)       |  acoth(x)       |  atanh(x)               |                                  
                   ||-  | -------- dx -  |  -------- dx -  |  -------- dx +  |  -------- dx +  |  -------- dx  otherwise |                                  
                   ||   |    x           |     x           |     x           |     x           |     x                   |                                  
                   ||   |                |                 |                 |                 |                         |                                  
                   ||  /                /                 /                 /                 /                          |                                  
                   \\                                                                                                    /

$$\int \frac{\log{\left(x \right)}}{x^{2} - 1}\, dx = C + \left(\begin{cases} - \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}\right) \log{\left(x \right)} - \begin{cases} - \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx & \text{for}\: x < -1 \\- \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x < 1 \\- \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{otherwise} \end{cases}$$

Simplificar

Respuesta [src]

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

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

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

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

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

Respuesta numérica [src]

1.23370055013617

1.23370055013617

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

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Integral de lnx/(x²-1) dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución