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Resolución de integrales/ lnx/(X^2+a^2)

Integral de lnx/(X^2+a^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo           
  /           
 |            
 |   log(x)   
 |  ------- dx
 |   2    2   
 |  x  + a    
 |            
/             
0
∫0∞log⁡(x)a2+x2 dx\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}}{a^{2} + x^{2}}\, dx 
Integral(log(x)/(x^2 + a^2), (x, 0, oo))
Respuesta [src] 
/                 /    pi*I\                /    3*pi*I\                                                                       
|                 |    ----|                |    ------|                                                                       
|                 |     2  |                |      2   |            /      3*pi*I\            /      pi*I\                     
|                 |   e    |                |   e      |            |      ------|            |      ----|                     
|  - 2*I*a*polylog|2, -----| + 2*I*a*polylog|2, -------|            |        2   |            |       2  |                     
|                 \     a  /                \      a   /   I*polylog\2, a*e      /   I*polylog\2, a*e    /                     
|- ----------------------------------------------------- + ----------------------- - ---------------------  for 2*|arg(a)| < pi
|                              2                                     2*a                      2*a                              
|                           4*a                                                                                                
|                                                                                                                              
<                                              oo                                                                              
|                                               /                                                                              
|                                              |                                                                               
|                                              |   log(x)                                                                      
|                                              |  ------- dx                                                     otherwise     
|                                              |   2    2                                                                      
|                                              |  a  + x                                                                       
|                                              |                                                                               
|                                             /                                                                                
|                                             0                                                                                
\
{−iLi⁡2(aeiπ2)2a+iLi⁡2(ae3iπ2)2a−−2iaLi⁡2(eiπ2a)+2iaLi⁡2(e3iπ2a)4a2for 2∣arg⁡(a)∣<π∫0∞log⁡(x)a2+x2 dxotherwise\begin{cases} - \frac{i \operatorname{Li}_{2}\left(a e^{\frac{i \pi}{2}}\right)}{2 a} + \frac{i \operatorname{Li}_{2}\left(a e^{\frac{3 i \pi}{2}}\right)}{2 a} - \frac{- 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{i \pi}{2}}}{a}\right) + 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{3 i \pi}{2}}}{a}\right)}{4 a^{2}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}}{a^{2} + x^{2}}\, dx & \text{otherwise} \end{cases} 
=
= 
/                 /    pi*I\                /    3*pi*I\                                                                       
|                 |    ----|                |    ------|                                                                       
|                 |     2  |                |      2   |            /      3*pi*I\            /      pi*I\                     
|                 |   e    |                |   e      |            |      ------|            |      ----|                     
|  - 2*I*a*polylog|2, -----| + 2*I*a*polylog|2, -------|            |        2   |            |       2  |                     
|                 \     a  /                \      a   /   I*polylog\2, a*e      /   I*polylog\2, a*e    /                     
|- ----------------------------------------------------- + ----------------------- - ---------------------  for 2*|arg(a)| < pi
|                              2                                     2*a                      2*a                              
|                           4*a                                                                                                
|                                                                                                                              
<                                              oo                                                                              
|                                               /                                                                              
|                                              |                                                                               
|                                              |   log(x)                                                                      
|                                              |  ------- dx                                                     otherwise     
|                                              |   2    2                                                                      
|                                              |  a  + x                                                                       
|                                              |                                                                               
|                                             /                                                                                
|                                             0                                                                                
\
{−iLi⁡2(aeiπ2)2a+iLi⁡2(ae3iπ2)2a−−2iaLi⁡2(eiπ2a)+2iaLi⁡2(e3iπ2a)4a2for 2∣arg⁡(a)∣<π∫0∞log⁡(x)a2+x2 dxotherwise\begin{cases} - \frac{i \operatorname{Li}_{2}\left(a e^{\frac{i \pi}{2}}\right)}{2 a} + \frac{i \operatorname{Li}_{2}\left(a e^{\frac{3 i \pi}{2}}\right)}{2 a} - \frac{- 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{i \pi}{2}}}{a}\right) + 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{3 i \pi}{2}}}{a}\right)}{4 a^{2}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}}{a^{2} + x^{2}}\, dx & \text{otherwise} \end{cases} 
Piecewise((-(-2*i*a*polylog(2, exp_polar(pi*i/2)/a) + 2*i*a*polylog(2, exp_polar(3*pi*i/2)/a))/(4*a^2) + i*polylog(2, a*exp_polar(3*pi*i/2))/(2*a) - i*polylog(2, a*exp_polar(pi*i/2))/(2*a), 2*Abs(arg(a)) < pi), (Integral(log(x)/(a^2 + x^2), (x, 0, oo)), True))

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

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