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Resolución de integrales/ 1-x^2/ ln(x)/ ln(x)/(1-x^2)

Integral de ln(x)/(1-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1          
  /          
 |           
 |  log(x)   
 |  ------ dx
 |       2   
 |  1 - x    
 |           
/            
1/2
∫121log⁡(x)1−x2 dx\int\limits_{\frac{1}{2}}^{1} \frac{\log{\left(x \right)}}{1 - x^{2}}\, dx 
Integral(log(x)/(1 - x^2), (x, 1/2, 1))
Respuesta (Indefinida) [src] 
                                                     //                                       /                                                            \
                                                     ||                                      |                                                             |
                                                     ||                                      | acoth(x)                                                    |
                                                     ||                                    - | -------- dx                                       for x < -1|
                                                     ||                                      |    x                                                        |
                                                     ||                                      |                                                             |
                                                     ||                                     /                                                              |
                                                     ||                                                                                                    |
                                                     ||                     -1                                 -1                                          |
                                                     ||                      /                 /                /                                          |
  /                                                  ||                     |                 |                |                                           |
 |                 //                2    \          ||                     |  acoth(x)       | atanh(x)       |  atanh(x)                                 |
 | log(x)          ||-acoth(x)  for x  > 1|          ||                  -  |  -------- dx -  | -------- dx +  |  -------- dx                    for x < 1 |
 | ------ dx = C - |<                     |*log(x) + |<                     |     x           |    x           |     x                                     |
 |      2          ||                2    |          ||                     |                 |                |                                           |
 | 1 - x           \\-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                   |
                                                     ||   |                |                 |                 |                 |                         |
                                                     ||  /                /                 /                 /                 /                          |
                                                     \\                                                                                                    /
∫log⁡(x)1−x2 dx=C−({−acoth⁡(x)for x2>1−atanh⁡(x)for x2<1)log⁡(x)+{−∫acoth⁡(x)x dxfor x<−1−∫−1acoth⁡(x)x dx−∫atanh⁡(x)x dx+∫−1atanh⁡(x)x dxfor x<1−∫acoth⁡(x)x dx−∫−1acoth⁡(x)x dx+∫1acoth⁡(x)x dx+∫−1atanh⁡(x)x dx−∫1atanh⁡(x)x dxotherwise\int \frac{\log{\left(x \right)}}{1 - x^{2}}\, 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
  |        2   
  |  -1 + x    
  |            
 /             
 1/2
−∫121log⁡(x)x2−1 dx- \int\limits_{\frac{1}{2}}^{1} \frac{\log{\left(x \right)}}{x^{2} - 1}\, dx 
=
= 
   1           
   /           
  |            
  |   log(x)   
- |  ------- dx
  |        2   
  |  -1 + x    
  |            
 /             
 1/2
−∫121log⁡(x)x2−1 dx- \int\limits_{\frac{1}{2}}^{1} \frac{\log{\left(x \right)}}{x^{2} - 1}\, dx 
-Integral(log(x)/(-1 + x^2), (x, 1/2, 1))
Respuesta numérica [src] 
-0.337623178232436
-0.337623178232436

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

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